1.1.8.1.
Hint.
Draw a rectangle that encompasses the entire shaded area, and one that is encompassed by the shaded area. The shaded area is no more than the area of the bigger rectangle, and no less than the area of the smaller rectangle.
Appendix E Hints for Exercises
1 Integration
1.1 Definition of the Integral
1.1.8 Exercises
1.1.8.2.
Hint.
We can improve on the method of Question 1 by using three rectangles that together encompass the shaded region, and three rectangles that together are encompassed by the shaded region.
1.1.8.3.
Hint.
Four rectangles suffice.
1.1.8.4.
Hint.
Try drawing a picture.
1.1.8.5.
Hint.
Try an oscillating function.
1.1.8.6.
Hint.
The ordering of the parts is intentional: each sum can be written by changing some small part of the sum before it.
1.1.8.7.
Hint.
If we raise \(-1\) to an even power, we get \(+1\text{,}\) and if we raise it to an odd power, we get \(-1\text{.}\)
1.1.8.8.
Hint.
Sometimes a little anti-simplification can make the pattern more clear.
- Re-write as \(\frac{1}{3}+\frac{3}{9}+\frac{5}{27}+\frac{7}{81}+\frac{9}{243}\text{.}\)
- Compare to the sum in the hint for (a).
- Re-write as \(1\cdot1000+2\cdot 100+3\cdot10+\frac{4}{1}+\frac{5}{10}+\frac{6}{100}+\frac{7}{1000}\text{.}\)
1.1.8.9.
Hint.
- (a), (b) These are geometric sums.
- (c) You can write this as three separate sums.
- (d) You can write this as two separate sums. Remember that \(e\) is a constant. Don’t be thrown off by the index being \(n\) instead of \(i\text{.}\)
1.1.8.10.
Hint.
- Write out the terms of the two sums.
- A change of index is an easier option than expanding the cubic.
- Which terms cancel?
- Remember \(2n+1\) is odd for every integer \(n\text{.}\) The index starts at \(n=2\text{,}\) not \(n=1\text{.}\)
1.1.8.11.
Hint.
Since the sum adds four pieces, there will be four rectangles. However, one might be extremely small.
1.1.8.12. (✳).
Hint.
Write out the general formula for the left Riemann sum from Definition 1.1.11 and choose \(a\text{,}\) \(b\) and \(n\) to make it match the given sum.
1.1.8.13.
Hint.
Since the sum runs from 1 to 3, there are three intervals. Suppose \(2 = \Delta x = \frac{b-a}{n}\text{.}\) You may assume the sum given is a right Riemann sum (as opposed to left or midpoint).
1.1.8.14.
Hint.
Let \(\Delta x = \dfrac{\pi}{20}\text{.}\) Then what is \(b-a\text{?}\)
1.1.8.15. (✳).
Hint.
Notice that the index starts at \(k=0\text{,}\) instead of \(k=1\text{.}\) Write out the given sum explicitly without using summation notation, and sketch where the rectangles would fall on a graph of \(y=f(x)\text{.}\)
Then try to identify \(b-a\text{,}\) and \(n\text{,}\) followed by “right”, “left”, or “midpoint”, and finally \(a\text{.}\)
1.1.8.16.
Hint.
The area is a triangle.
1.1.8.17.
Hint.
There is one triangle of positive area, and one of negative area.
1.1.8.18. (✳).
Hint.
Review Definition 1.1.11.
1.1.8.20. (✳).
Hint.
You’ll want the limit as \(n\) goes to infinity of a sum with \(n\) terms. If you’re having a hard time coming up with the sum in terms of \(n\text{,}\) try writing a sum with a finite number of terms of your choosing. Then, think about how that sum would change if it had \(n\) terms.
1.1.8.21. (✳).
Hint.
The main step is to express the given sum as the right Riemann sum,
\begin{equation*}
\sum_{i=1}^{n} f(a+i\De x)\Delta x.
\end{equation*}
Don’t be afraid to guess \(\De x\) and \(f(x\)) (review Definition 1.1.11). Then write out explicitly \(\sum\limits_{i=1}^{n} f(a+i\De x)\Delta x\) with your guess substituted in, and compare the result with the given sum. Adjust your guess if they don’t match.
1.1.8.22. (✳).
Hint.
The main step is to express the given sum as the right Riemann sum \(\sum\limits_{k=1}^{n} f(a+k\De x)\Delta x\text{.}\) Don’t be afraid to guess \(\De x\) and \(f(x\)) (review Definition 1.1.11). Then write out explicitly \(\sum\limits_{k=1}^{n} f(a+k\De x)\Delta x\) with your guess substituted in, and compare the result with the given sum. Adjust your guess if they don’t match.
1.1.8.23. (✳).
Hint.
The main step is to express the given sum in the form \(\sum_{i=1}^{n} f(x_i^*)\Delta x\text{.}\) Don’t be afraid to guess \(\De x\text{,}\) \(x_i^*\) (for either a left or a right or a midpoint sum — review Definition 1.1.11) and \(f(x\)). Then write out explicitly \(\sum_{i=1}^{n} f(x_i^*)\Delta x\) with your guess substituted in, and compare the result with the given sum. Adjust your guess if they don’t match.
1.1.8.24. (✳).
Hint.
The main step is to express the given sum in the form \(\sum\limits_{i=1}^{n} f(x_i^*)\Delta x\text{.}\) Don’t be afraid to guess \(\De x\text{,}\) \(x_i^*\) (probably, based on the symbol \(R_n\text{,}\) assuming we have a right Riemann sum — review Definition 1.1.11) and \(f(x\)). Then write out explicitly \(\sum\limits_{i=1}^{n} f(x_i^*)\Delta x\) with your guess substituted in, and compare the result with the given sum. Adjust your guess if they don’t match.
1.1.8.25. (✳).
Hint.
Try several different choices of \(\De x\) and \(x_i^*\text{.}\)
1.1.8.26.
Hint.
Let \(x=r^3\text{,}\) and re--write the sum in terms of \(x\text{.}\)
1.1.8.27.
Hint.
Note the sum does not start at \(r^0=1\text{.}\)
1.1.8.28. (✳).
Hint.
Draw a picture. See Example 1.1.15.
1.1.8.29.
Hint.
Draw a picture. Remember \(|x| = \left\{\begin{array}{rc}x&x\ge 0\\-x&x \lt 0\end{array}\right.\text{.}\)
1.1.8.30.
Hint.
Draw a picture: the area we want is a trapezoid. If you don’t remember a formula for the area of a trapezoid, think of it as the difference of two triangles.
1.1.8.31.
Hint.
You can draw a very similar picture to Question 30, but remember the areas are negative.
1.1.8.32.
Hint.
If \(y=\sqrt{16-x^2}\text{,}\) then \(y\) is nonnegative, and \(y^2+x^2=16\text{.}\)
1.1.8.33. (✳).
Hint.
Sketch the graph of \(f(x)\text{.}\)
1.1.8.34. (✳).
Hint.
At which time in the interval, for example, \(0\le t\le 0.5\text{,}\) is the car moving the fastest?
1.1.8.35.
Hint.
What are the possible speeds the car could have reached at time \(t=0.25\text{?}\)
1.1.8.36.
Hint.
You need to know the speed of the plane at the midpoints of your intervals, so (for example) noon to 1pm is not one of your intervals.
1.1.8.37. (✳).
Hint.
Sure looks like a Riemann sum.
1.1.8.38. (✳).
Hint.
For part (b): don’t panic! Just take it one step at a time. The first step is to write down the Riemann sum. The second step is to evaluate the sum, using the given identity. The third step is to evaluate the limit \(n\rightarrow\infty\text{.}\)
1.1.8.39. (✳).
Hint.
The first step is to write down the Riemann sum. The second step is to evaluate the sum, using the given formulas. The third step is to evaluate the limit as \(n\rightarrow\infty\text{.}\)
1.1.8.40. (✳).
Hint.
The first step is to write down the Riemann sum. The second step is to evaluate the sum, using the given formulas. The third step is to evaluate the limit \(n\rightarrow\infty\text{.}\)
1.1.8.41. (✳).
Hint.
You’ve probably seen this hint before. It is worth repeating. Don’t panic! Just take it one step at a time. The first step is to write down the Riemann sum. The second step is to evaluate the sum, using the given formula. The third step is to evaluate the limit \(n\rightarrow\infty\text{.}\)
1.1.8.42.
Hint.
Using the definition of a right Riemann sum, we can come up with an expression for \(f(-5+10i)\text{.}\) In order to find \(f(x)\text{,}\) set \(x=-5+10i\text{.}\)
1.1.8.43.
Hint.
Recall that for a positive constant \(a\text{,}\) \(\diff{}{x}\left\{a^x\right\} = a^x \log a\text{,}\) where \(\log a\) is the natural logarithm (base \(e\)) of \(a\text{.}\)
1.1.8.44.
Hint.
Part (a) follows the same pattern as Question 43--there’s just a little more algebra involved, since our lower limit of integration is not 0.
1.1.8.45.
Hint.
Your area can be divided into a section of a circle and a triangle. Then you can use geometry to find the area of each piece.
1.1.8.46.
Hint.
- The difference between the upper and lower bounds is the area that is outside of the smaller rectangles but inside the larger rectangles. Drawing both sets of rectangles on one picture might make things clearer. Look for an easy way to compute the area you want.
- Use your answer from Part (a). Your answer will depend on \(f\text{,}\) \(a\text{,}\) and \(b\text{.}\)
1.1.8.47.
Hint.
Since \(f(x)\) is linear, there exist real numbers \(m\) and \(c\) such that \(f(x)=mx+c\text{.}\) It’s a little easier to first look at a single triangle from each sum, rather than the sums in their entirety.
1.2 Basic properties of the definite integral
1.2.3 Exercises
1.2.3.1.
Hint.
- What is the length of this figure?
- Think about cutting the area into two pieces vertically.
- Think about cutting the area into two pieces another way.
1.2.3.2.
Hint.
Use the identity \(\int\limits_a^b f(x)\dee{x} =
\int\limits_a^c f(x)\dee{x}+
\int\limits_c^b f(x)\dee{x}\text{.}\)
1.2.3.4.
Hint.
Note that the limits of the integral given are in the opposite order from what we might expect: the smaller number is the top limit of integration.
Recall \(\De x = \frac{b-a}{n}\text{.}\)
1.2.3.5. (✳).
Hint.
Split the “target integral” up into pieces that can be evaluated using the given integrals.
1.2.3.6. (✳).
Hint.
Split the “target integral” up into pieces that can be evaluated using the given integrals.
1.2.3.7. (✳).
Hint.
Split the “target integral” up into pieces that can be evaluated using the given integrals.
1.2.3.8.
Hint.
For part (a), use the symmetry of the integrand. For part (b), the area \(\int \limits_{0}^1 \sqrt{1-x^2}\dee{x}\) is easy to find--how is this useful to you?
1.2.3.9. (✳).
1.2.3.10.
Hint.
Use symmetry.
1.2.3.11.
Hint.
Check Theorem 1.2.12.
1.2.3.12. (✳).
Hint.
Split the integral into a sum of two integrals. Interpret each geometrically.
1.2.3.13. (✳).
Hint.
Hmmmm. Looks like a complicated integral. It’s probably a trick question. Check for symmetries.
1.2.3.14. (✳).
Hint.
Check for symmetries again.
1.2.3.15.
Hint.
What does the integrand look like to the left and right of \(x=3\text{?}\)
1.2.3.16.
Hint.
In part (b), you’ll have to factor a constant out through a square root. Remember the upper half of a circle looks like \(\sqrt{r^2-x^2}\text{.}\)
1.2.3.17.
Hint.
For two functions \(f(x)\) and \(g(x)\text{,}\) define \(h(x)=f(x)\cdot g(x)\text{.}\) If \(h(-x)=h(x)\text{,}\) then the product is even; if \(h(-x)=-h(x)\text{,}\) then the product is odd.
The table will not be the same as if we were multiplying even and odd numbers.
1.2.3.18.
Hint.
Note \(f(0)=f(-0)\text{.}\)
1.2.3.19.
Hint.
If \(f(x)\) is even and odd, then \(f(x)=-f(x)\) for every \(x\text{.}\)
1.2.3.20.
Hint.
Think about mirroring a function across an axis. What does this do to the slope?
1.3 The Fundamental Theorem of Calculus
1.3.2 Exercises
1.3.2.2. (✳).
Hint.
First find the general antiderivative by guessing and checking.
1.3.2.3. (✳).
Hint.
Be careful. Two of these make no sense at all.
1.3.2.4.
Hint.
Check by differentiating.
1.3.2.5.
Hint.
Check by differentiating.
1.3.2.6.
Hint.
Use the Fundamental Theorem of Calculus Part 1.
1.3.2.7.
Hint.
Use the Fundamental Theorem of Calculus, Part 1.
1.3.2.8.
Hint.
You already know that \(F(x)\) is an antiderivative of \(f(x)\text{.}\)
1.3.2.9.
Hint.
(a) Recall \(\diff{}{x}\{\arccos x\} = \frac{-1}{\sqrt{1-x^2}}\text{.}\)
(b) All antiderivatives of \(\sqrt{1-x^2}\) differ from one another by a constant. You already know one antiderivative.
1.3.2.10.
Hint.
In order to apply the Fundamental Theorem of Calculus Part 2, the integrand must be continuous over the interval of integration.
1.3.2.11.
Hint.
Use the definition of \(F(x)\) as an area.
1.3.2.12.
Hint.
\(F(x)\) represents net signed area.
1.3.2.13.
Hint.
1.3.2.14.
Hint.
Using the definition of the derivative, \(F'(x) = \displaystyle\lim_{h \to 0}\dfrac{F(x+h)-F(x)}{h}\text{.}\)
The area of a trapezoid with base \(b\) and heights \(h_1\) and \(h_2\) is \(\frac{1}{2}b(h_1+h_2)\text{.}\)
1.3.2.15.
Hint.
There is only one!
1.3.2.16.
Hint.
If \(\diff{}{x}\{F(x)\}=f(x)\text{,}\) that tells us \(\int f(x)\dee{x} = F(x)+C\text{.}\)
1.3.2.17.
Hint.
When you’re differentiating, you can leave the \(e^x\) factored out.
1.3.2.18.
Hint.
After differentiation, you can simplify pretty far. Keep at it!
1.3.2.19.
Hint.
This derivative also simplifies considerably. You might need to add fractions by finding a common denominator.
1.3.2.20. (✳).
Hint.
Guess a function whose derivative is the integrand, then use the Fundamental Theorem of Calculus Part 2.
1.3.2.21. (✳).
Hint.
Split the given integral up into two integrals.
1.3.2.22.
Hint.
The integrand is similar to \(\dfrac{1}{1+x^2}\text{,}\) so something with arctangent seems in order.
1.3.2.23.
Hint.
The integrand is similar to \(\dfrac{1}{\sqrt{1-x^2}}\text{,}\) so factoring out \(\sqrt{2}\) from the denominator will make it look like some flavour of arcsine.
1.3.2.24.
Hint.
We know how to antidifferentiate \(\sec^2 x\text{,}\) and there is an identity linking \(\sec^2 x\) with \(\tan^2 x\text{.}\)
1.3.2.25.
Hint.
Recall \(2\sin x \cos x = \sin(2x)\text{.}\)
1.3.2.26.
Hint.
\(\cos^2 x = \dfrac{1+\cos(2x)}{2}\)
1.3.2.28. (✳).
Hint.
There is a good way to test where a function is increasing, decreasing, or constant, that also has something to do with topic of this section.
1.3.2.29. (✳).
Hint.
See Example 1.3.5.
1.3.2.30. (✳).
Hint.
See Example 1.3.5.
1.3.2.31. (✳).
Hint.
See Example 1.3.5.
1.3.2.32. (✳).
Hint.
See Example 1.3.5.
1.3.2.33. (✳).
Hint.
See Example 1.3.6.
1.3.2.34. (✳).
Hint.
Apply \(\diff{}{x}\) to both sides.
1.3.2.35. (✳).
Hint.
What is the title of this section?
1.3.2.36. (✳).
Hint.
See Example 1.3.6.
1.3.2.37. (✳).
Hint.
See Example 1.3.6.
1.3.2.38. (✳).
Hint.
See Example 1.3.6.
1.3.2.39. (✳).
Hint.
See Example 1.3.6.
1.3.2.40. (✳).
Hint.
Split up the domain of integration.
1.3.2.41. (✳).
Hint.
It is possible to guess an antiderivative for \(f'(x) f''(x)\) that is expressed in terms of \(f'(x)\text{.}\)
1.3.2.42. (✳).
Hint.
When does the car stop? What is the relation between velocity and distance travelled?
1.3.2.43. (✳).
Hint.
See Example 1.3.5. For the absolute maximum part of the question, study the sign of \(f'(x)\text{.}\)
1.3.2.44. (✳).
Hint.
See Example 1.3.5. For the “minimum value” part of the question, study the sign of \(f'(x)\text{.}\)
1.3.2.45. (✳).
Hint.
See Example 1.3.5. For the “maximum” part of the question, study the sign of \(F'(x)\text{.}\)
1.3.2.46. (✳).
1.3.2.47. (✳).
1.3.2.48.
Hint.
Carefully check the Fundamental Theorem of Calculus: as written, it only applies directly to \(F(x)\) when \(x\ge0\text{.}\)
Is \(F(x)\) even or odd?
1.3.2.49. (✳).
Hint.
In general, the equation of the tangent line to the graph of \(y=f(x)\) at \(x=a\) is \(y=f(a) + f'(a)\,(x-a)\text{.}\)
1.3.2.50.
Hint.
Recall \(\tan^2x+1=\sec^2 x\text{.}\)
1.3.2.51.
Hint.
Since the integration is with respect to \(t\text{,}\) the \(x^3\) term can be moved outside the integral.
1.3.2.52.
Hint.
Remember that antiderivatives may have a constant term.
1.4 Substitution
1.4.2 Exercises
1.4.2.1.
Hint.
One is true, the other false.
1.4.2.2.
Hint.
You can check whether the final answer is correct by differentiating.
1.4.2.3.
Hint.
Check the limits.
1.4.2.4.
Hint.
Check every step. Do they all make sense?
1.4.2.6.
Hint.
What is \(\diff{}{x}\{f(g(x))\}\text{?}\)
1.4.2.7. (✳).
Hint.
What is the derivative of the argument of the cosine?
1.4.2.8. (✳).
Hint.
What is the title of the current section?
1.4.2.9. (✳).
Hint.
What is the derivative of \(x^3+1\text{?}\)
1.4.2.10. (✳).
Hint.
What is the derivative of \(\log x\text{?}\)
1.4.2.11. (✳).
Hint.
What is the derivative of \(1+\sin x\text{?}\)
1.4.2.12. (✳).
Hint.
\(\cos x\) is the derivative of what?
1.4.2.13. (✳).
Hint.
What is the derivative of the exponent?
1.4.2.14. (✳).
Hint.
What is the derivative of the argument of the square root?
1.4.2.15.
Hint.
What is \(\diff{}{x}\left\{\sqrt{\log x}\right\}\text{?}\)
1.4.2.16. (✳).
Hint.
There is a short, slightly sneaky method — guess an antiderivative — and a really short, still-more-sneaky method.
1.4.2.17. (✳).
1.4.2.18.
Hint.
If \(w=u^2+1\text{,}\) then \(u^2=w-1\text{.}\)
1.4.2.19.
Hint.
Using a trigonometric identity, this is similar (though not identical) to \(\int \tan \theta \cdot \sec^2 \theta\dee{\theta}\text{.}\)
1.4.2.20.
Hint.
If you multiply the top and the bottom by \(e^x\text{,}\) what does this look like the antiderivative of?
1.4.2.21.
Hint.
You know methods other than substitution to evaluate definite integrals.
1.4.2.22.
Hint.
\(\tan x = \dfrac{\sin x}{\cos x}\)
1.4.2.23. (✳).
1.4.2.24. (✳).
1.4.2.25.
Hint.
Find the right Riemann sum for both definite integrals.
1.5 Area between curves
1.5.2 Exercises
1.5.2.1.
Hint.
When we say “area between,” we want positive area, not signed area.
1.5.2.2.
Hint.
We’re taking rectangles that reach from one function to the other.
1.5.2.3. (✳).
Hint.
Draw a sketch first.
1.5.2.4. (✳).
Hint.
Draw a sketch first.
1.5.2.5. (✳).
Hint.
You can probably find the intersections by inspection.
1.5.2.6. (✳).
Hint.
To find the intersection, plug \(x=4y^2\) into the equation \(x+12y+5=0\text{.}\)
1.5.2.7. (✳).
Hint.
If the bottom function is the \(x\)-axis, this is a familiar question.
1.5.2.8. (✳).
Hint.
Part of the job is to determine whether \(y=x\) lies above or below \(y=3x-x^2\text{.}\)
1.5.2.9. (✳).
Hint.
Guess the intersection points by trying small integers.
1.5.2.10. (✳).
Hint.
Draw a sketch first. You can also exploit a symmetry of the region to simplify your solution.
1.5.2.11. (✳).
Hint.
Figure out where the two curves cross. To determine which curve is above the other, try evaluating \(f(x)\) and \(g(x)\) for some simple value of \(x\text{.}\) Alternatively, consider \(x\) very close to zero.
1.5.2.12. (✳).
Hint.
Think about whether it will easier to use vertical strips or horizontal strips.
1.5.2.13.
Hint.
Writing an integral for this is nasty. How can you avoid it?
1.5.2.14. (✳).
Hint.
You are asked for the area, not the signed area. Be very careful about signs.
1.5.2.15. (✳).
Hint.
You are asked for the area, not the signed area. Draw a sketch of the region and be very careful about signs.
1.5.2.16. (✳).
Hint.
You have to determine whether
- the curve \(y = f(x) = x \sqrt{25-x^2}\) lies above the line \(y=g(x)=3x\) for all \(0\le x\le 4\) or
- the curve \(y = f(x)\) lies below the line \(y=g(x)\) for all \(0\le x\le 4\) or
- \(y=f(x)\) and \(y=g(x)\) cross somewhere between \(x=0\) and \(x=4\text{.}\)
One way to do so is to study the sign of \(f(x)-g(x) = x\big(\sqrt{25-x^2}-3\big)\text{.}\)
1.5.2.17.
Hint.
Flex those geometry muscles.
1.5.2.18.
Hint.
These two functions have three points of intersection. This question is slightly messy, but uses the same concepts we’ve been practicing so far.
1.6 Volumes
1.6.2 Exercises
1.6.2.1.
Hint.
The horizontal cross-sections were discussed in Example 1.6.1.
1.6.2.2.
Hint.
What are the dimensions of the cross-sections?
1.6.2.3.
Hint.
There are two different kinds of washers.
1.6.2.4. (✳).
Hint.
Draw sketches. The mechanically easiest way to answer part (b) uses the method of cylindrical shells, which is in the optional section 1.6. The method of washers also works, but requires you to have more patience and also to have a good idea what the specified region looks like. Look at your sketch very careful when identifying the ends of your horizontal strips.
1.6.2.5. (✳).
Hint.
Draw sketchs.
1.6.2.6. (✳).
Hint.
Draw a sketch.
1.6.2.7.
Hint.
If you take horizontal slices (parallel to one face), they will all be equilateral triangles.
Be careful not to confuse the height of a triangle with the height of the tetrahedron.
1.6.2.8. (✳).
Hint.
Sketch the region.
1.6.2.9. (✳).
Hint.
Sketch the region first.
1.6.2.10. (✳).
Hint.
You can save yourself quite a bit of work by interpreting the integral as the area of a known geometric figure.
1.6.2.11. (✳).
Hint.
See Example 1.6.3.
1.6.2.12. (✳).
Hint.
See Example 1.6.5.
1.6.2.13. (✳).
Hint.
Sketch the region. To find where the curves intersect, look at where \(\cos(\frac x2)\) and \(x^2 - \pi^2\) both have roots.
1.6.2.14. (✳).
Hint.
See Example 1.6.6.
1.6.2.15. (✳).
Hint.
See Example 1.6.6. Imagine cross-sections with shadow parallel to the \(y\)-axis, sticking straight out of the \(xy\)-plane.
1.6.2.16. (✳).
Hint.
See Example 1.6.1.
1.6.2.17.
Hint.
(a) Don’t be put off by phrases like “rotating an ellipse about its minor axis.” This is the same kind of volume you’ve been calculating all section.
(b) Hopefully, you sketched the ellipse in part (a). What was its smallest radius? Its largest? These correspond to the polar and equitorial radii, respectively.
(c) Combine your answers from (a) and (b).
(d) Remember that the absolute error is the absolute difference of your two results--that is, you subtract them and take the absolute value. The relative error is the absolute error divided by the actual value (which we’re taking, for our purposes, to be your answer from (c)). When you take the relative error, lots of terms will cancel, so it’s easiest to not use a calculator till the end.
1.6.2.18. (✳).
Hint.
To find the points of intersection, set \(4-(x-1)^2=x+1\text{.}\)
1.6.2.19. (✳).
Hint.
You can somewhat simplify your calculations in part (a) (but not part (b)) by using the fact that \(\cR\) is symmetric about the line \(y=x\text{.}\)
When you’re solving an equation for \(x\text{,}\) be careful about your signs: \(x-1\) is negative.
1.6.2.20. (✳).
Hint.
The mechanically easiest way to answer part (b) uses the method of cylindrical shells, which we have not covered. The method of washers also works, but requires you have enough patience and also to have a good idea what \(\cR\) looks like. So it is crucial to first sketch \(\cR\text{.}\) Then be very careful in identifying the left end of your horizontal strips.
1.6.2.21. (✳).
1.6.2.22.
Hint.
You can use ideas from this section to answer the question. If you take a very thin slice of the column, the density is almost constant, so you can find the mass. Then you can add up all your little slices. It’s the same idea as volume, only applied to mass.
Do be careful about units: in the problem statement, some are given in metres, others in kilometres.
If you’re having a hard time with the antiderivative, try writing the exponential function with base \(e\text{.}\) Remember \(2 = e^{\log 2}\text{.}\)
1.7 Integration by parts
1.7.2 Exercises
1.7.2.1.
1.7.2.2.
Hint.
Remember our rule: \(\int u \dee{v} = uv - \int v \dee{u}\text{.}\) So, we take \(u\) and use it to make \(\dee{u}\text{,}\) and we take \(\dee{v}\) and use it to make \(v\text{.}\)
1.7.2.3.
Hint.
According to the quotient rule,
\begin{equation*}
\diff{}{x}\left\{\dfrac{f(x)}{g(x)}\right\} = \frac{g(x)f'(x)-f(x)g'(x)}{g^2(x)}.
\end{equation*}
Antidifferentiate both sides of the equation, then solve for the expression in the question.
1.7.2.4.
Hint.
Remember all the antiderivatives differ only by a constant, so you can write them all as \(v(x)+C\) for some \(C\text{.}\)
1.7.2.5.
Hint.
What integral do you have to evaluate, after you plug in your choices to the integration by parts formula?
1.7.2.6. (✳).
Hint.
You’ll probably want to use integration by parts. (It’s the title of the section, after all). You’ll break the integrand into two parts, integrate one, and differentiate the other. Would you rather integrate \(\log x\text{,}\) or differentiate it?
1.7.2.7. (✳).
Hint.
This problem is similar to Question 6.
1.7.2.8. (✳).
Hint.
Example 1.7.5 shows you how to find the antiderivative. Then the Fundamental Theorem of Calculus Part 2 gives you the definite integral.
1.7.2.9. (✳).
Hint.
Compare to Question 8. Try to do this one all the way through without peeking at another solution!
1.7.2.10.
Hint.
If at first you don’t succeed, try using integration by parts a few times in a row. Eventually, one part will go away.
1.7.2.11.
Hint.
Similarly to Question 10, look for a way to use integration by parts a few times to simplify the integrand until it is antidifferentiatable.
1.7.2.12.
Hint.
Use integration by parts twice to get an integral with only a trigonometric function in it.
1.7.2.13.
Hint.
If you let \(u=\log t\) in the integration by parts, then \(\dee{u}\) works quite nicely with the rest of the integrand.
1.7.2.14.
Hint.
Those square roots are a little disconcerting-- get rid of them with a substitution.
1.7.2.15.
Hint.
This can be solved using the same ideas as Example 1.7.8 in your text.
1.7.2.16.
Hint.
Not every integral should be evaluated using integration by parts.
1.7.2.17. (✳).
Hint.
You know, or can easily look up, the derivative of arccosine. You can use a similar trick as the book did when antidifferentiating other inverse trigonometric functions in Example 1.7.9.
1.7.2.18. (✳).
Hint.
After integrating by parts, do some algebraic manipulation to the integral until it’s clear how to evaluate it.
1.7.2.19.
Hint.
After integration by parts, use a substitution.
1.7.2.20.
Hint.
This example is similar to Example 1.7.10 in the text. The functions \(e^{x/2}\) and \(\cos(2x)\) both do not substantially alter when we differentiate or antidifferentiate them. If we use integration by parts twice, we’ll end up with an expression that includes our original integral. Then we can just solve for the original integral in the equation, without actually integrating.
1.7.2.21.
Hint.
This looks a bit like a substitution problem, because we have an “inside function.”
It might help to review Example 1.7.11.
1.7.2.22.
Hint.
Start by simplifying.
1.7.2.23.
Hint.
\(\sin(2x) = 2\sin x \cos x\)
1.7.2.24.
Hint.
What is the derivative of \(x e^{-x}\text{?}\)
1.7.2.25. (✳).
Hint.
You’ll want to do an integration by parts for (a)--check the end result to get a guess as to what your parts should be. A trig identity and some amount of algebraic manipulation will be necessary to get the final form.
1.7.2.26. (✳).
1.7.2.27. (✳).
Hint.
Your integral can be broken into two integrals, which yield to two different integration methods.
1.7.2.28. (✳).
Hint.
Think, first, about how to get rid of the square root in the argument of \(f''\text{,}\) and, second, how to convert \(f''\) into \(f'\text{.}\) Note that you are told that \(f'(2) = 4\) and \(f(0) = 1\text{,}\) \(f(2) = 3\text{.}\)
1.7.2.29.
Hint.
Interpret the limit as a right Riemann sum.
1.8 Trigonometric Integrals
1.8.4 Exercises
1.8.4.1.
Hint.
Go ahead and try it!
1.8.4.2.
Hint.
Use the substitution \(u=\sec x\text{.}\)
1.8.4.3.
Hint.
Divide both sides of the second identity by \(\cos^2 x\text{.}\)
1.8.4.4. (✳).
Hint.
See Example 1.8.6. Note that the power of cosine is odd, and the power of sine is even (it’s zero).
1.8.4.5. (✳).
Hint.
See Example 1.8.7. All you need is a helpful trig identity.
1.8.4.6. (✳).
Hint.
The power of cosine is odd, so we can reserve one cosine for \(\dee{u}\text{,}\) and turn the rest into sines using the identity \(\sin^2 x + \cos^2 x =1\text{.}\)
1.8.4.7.
Hint.
Since the power of sine is odd (and positive), we can reserve one sine for \(\dee{u}\text{,}\) and turn the rest into cosines using the identity \(\sin^2 + \cos^2 x =1\text{.}\)
1.8.4.8.
Hint.
When we have even powers of sine and cosine both, we use the identities in the last two lines of Equation 1.8.3.
1.8.4.9.
Hint.
Since the power of sine is odd, you can use the substitution \(u=\cos x\text{.}\)
1.8.4.10.
Hint.
Which substitution will work better: \(u=\sin x\text{,}\) or \(u=\cos x\text{?}\)
1.8.4.11.
Hint.
Try a substitution.
1.8.4.12. (✳).
Hint.
For practice, try doing this in two ways, with different substitutions.
1.8.4.13. (✳).
Hint.
A substitution will work. See Example 1.8.14 for a template for integrands with even powers of secant.
1.8.4.14.
Hint.
Try the substitution \(u=\sec x\text{.}\)
1.8.4.15.
Hint.
Compare to Question 14.
1.8.4.16.
Hint.
What is the derivative of tangent?
1.8.4.17.
Hint.
Don’t be scared off by the non-integer power of secant. You can still use the strategies in the notes for an odd power of tangent.
1.8.4.18.
Hint.
Since there are no secants in the problem, it’s difficult to use the substitution \(u=\sec x\) that we’ve enjoyed in the past. Example 1.8.12 in the text provides a template for antidifferentiating an odd power of tangent.
1.8.4.19.
Hint.
Integrating even powers of tangent is surprisingly different from integrating odd powers of tangent. You’ll want to use the identity \(\tan^2x = \sec^2 x -1\text{,}\) then use the substitution \(u=\tan x\text{,}\) \(\dee{u}=\sec^2 x\dee{x}\) on (perhaps only a part of) the resulting integral. Example 1.8.16 show you how this can be accomplished.
1.8.4.20.
Hint.
Since there is an even power of secant in the integrand, we can use the substitution \(u=\tan x\text{.}\)
1.8.4.21.
Hint.
How have we handled integration in the past that involved an odd power of tangent?
1.8.4.22.
Hint.
Remember \(e\) is some constant. What are our strategies when the power of secant is even and positive? We’ve seen one such substitution in Example 1.8.15.
1.8.4.23. (✳).
Hint.
See Example 1.8.16 for a strategy for integrating powers of tangent.
1.8.4.24.
Hint.
Write \(\tan x = \dfrac{\sin x}{\cos x}\text{.}\)
1.8.4.25.
Hint.
\(\dfrac{1}{\cos \theta} = \sec \theta\)
1.8.4.26.
Hint.
\(\cot x = \dfrac{\cos x}{\sin x}\)
1.8.4.27.
Hint.
Try substituting.
1.8.4.28.
Hint.
To deal with the “inside function,” start with a substitution.
1.8.4.29.
Hint.
Try an integration by parts.
1.9 Trigonometric Substitution
1.9.2 Exercises
1.9.2.1. (✳).
Hint.
The beginning of this section has a template for choosing a substitution. Your goal is to use a trig identity to turn the argument of the square root into a perfect square, so you can cancel \(\sqrt{(\mbox{something})^2}=|\mbox{something}|\text{.}\)
1.9.2.2.
Hint.
You want to do the same thing you did in Question 1, but you’ll have to complete the square first.
1.9.2.3.
Hint.
Since \(\theta\) is acute, you can draw it as an angle of a right triangle. The given information will let you label two sides of the triangle, and the Pythagorean Theorem will lead you to the third.
1.9.2.4.
Hint.
You can draw a right triangle with angle \(\theta\text{,}\) and use the given information to label two of the sides. The Pythagorean Theorem gives you the third side.
1.9.2.5. (✳).
1.9.2.6. (✳).
Hint.
As in Question 1, choose an appropriate substitution. Your answer will be a number, so as long as you change your limits of integration when you substitute, you don’t need to bother changing the antiderivative back into the original variable \(x\text{.}\) However, you might want to use the techniques of Question 4 to simplify your final answer.
1.9.2.7. (✳).
Hint.
Question 1 guides the way to finding the appropriate substitution. Since the integral is definite, your final answer will be a number. Your limits of integration should be common reference angles.
1.9.2.8. (✳).
1.9.2.9.
Hint.
A trig substitution is not the easiest path.
1.9.2.10. (✳).
Hint.
To antidifferentiate, change your trig functions into sines and cosines.
1.9.2.11. (✳).
Hint.
The integrand should simplify quite far after your substitution.
1.9.2.12. (✳).
Hint.
In part (a) you are asked to integrate an even power of \(\cos x\text{.}\) For part (b) you can use a trigonometric substitution to reduce the integral of part (b) almost to the integral of part (a).
1.9.2.13.
Hint.
What is the symmetry of the integrand?
1.9.2.14. (✳).
Hint.
See Example 1.9.3.
1.9.2.15. (✳).
Hint.
To integrate an even power of tangent, use the identity \(\tan^2 x = \sec^2 x - 1\text{.}\)
1.9.2.16.
Hint.
A trig substitution is not the easiest path.
1.9.2.17. (✳).
Hint.
Complete the square. Your final answer will have an inverse trig function in it.
1.9.2.18.
Hint.
To antidifferentiate even powers of cosine, use the formula \(\cos^2\theta = \frac{1}{2}(1+\cos(2\theta))\text{.}\) Then, remember \(\sin(2\theta)=2\sin\theta\cos\theta\text{.}\)
1.9.2.19.
Hint.
After substituting, use the identity \(\tan^2 x = \sec^2 x - 1\) more than once.
Remember \(\displaystyle\int \sec x \dee{x} = \log \big|\sec x + \tan x \big|+C\text{.}\)
1.9.2.20.
Hint.
There’s no square root, but we can still make use of the substitution \(x=\tan\theta\text{.}\)
1.9.2.21.
Hint.
You’ll probably want to use the identity \(\tan^2\theta+1=\sec^2\theta\) more than once.
1.9.2.22.
Hint.
Complete the square — refer to Question 2 if you want a refresher. The constants aren’t pretty, but don’t let them scare you.
1.9.2.23.
Hint.
After substituting, use the identity \(\sec^2 u = \tan^2 u +1\text{.}\) It might help to break the integral into a few pieces.
1.9.2.24.
Hint.
Make use of symmetry, and integrate with respect to \(y\) (rather than \(x\)).
1.9.2.25.
Hint.
Use the symmetry of the function to re-write your integrals without an absolute value.
1.9.2.26.
Hint.
Think of \(e^x\) as \(\left(e^{x/2}\right)^2\text{,}\) and use a trig substitution. Then, use the identity \(\sec^2 \theta = \tan^2 \theta +1\text{.}\)
1.9.2.27.
Hint.
- Use logarithm rules to simplify first.
- Think about domains.
-
What went wrong in part (b)? At what point in the work was that problem introduced?There is a subtle but important point mentioned in the introductory text to Section 1.9 that may help you make sense of things.
1.9.2.28.
Hint.
Consider the ranges of the inverse trigonometric functions. For (c), also consider the domain of \(\sqrt{x^2-a^2}\text{.}\)
1.10 Partial Fractions
1.10.4 Exercises
1.10.4.1.
Hint.
If a quadratic function can be factored as \((ax+b)(cx+d)\) for some constants \(a,b,c,d\text{,}\) then it has roots \(-\frac{b}{a}\) and \(-\frac{d}{c}\text{.}\)
1.10.4.2. (✳).
1.10.4.3. (✳).
Hint.
Review Example 1.10.1. Is the “Algebraic Method” or the “Sneaky Method” going to be easier?
1.10.4.4.
Hint.
For each part, use long division as in Example 1.10.4.
1.10.4.5.
Hint.
(a) Look for a pattern you can exploit to factor out a linear term.
(b) If you set \(y=x^2\text{,}\) this is quadratic. Remember \((x^2-a)= (x+\sqrt{a})(x-\sqrt{a})\) as long as \(a\) is positive
(c),(d) Look for integer roots, then use long division.
1.10.4.6.
Hint.
Why do we do partial fraction decomposition at all?
1.10.4.7. (✳).
Hint.
What is the title of this section?
1.10.4.8. (✳).
Hint.
You can save yourself some work in developing your partial fraction decomposition by renaming \(x^2\) to \(y\) and comparing the result with Question 7.
1.10.4.9. (✳).
Hint.
Review Steps 3 (particularly the “Sneaky Method”) and 4 of Example 1.10.3.
1.10.4.10. (✳).
Hint.
Review Steps 3 (particularly the “Sneaky Method”) and 4 of Example 1.10.3. Remember \(\diff{}{x}\{\arctan x\} = \frac{1}{1+x^2}\text{.}\)
1.10.4.11. (✳).
Hint.
Fill in the blank: the integrand is a \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \) function.
1.10.4.12. (✳).
Hint.
The integrand is yet another \(\ \ \ \ \ \ \ \ \ \ \ \ \ \) function.
1.10.4.13.
Hint.
Since the degree of the numerator is the same as the degree of the denominator, we can’t do our partial fraction decomposition before we simplify the integrand.
1.10.4.14.
Hint.
The degree of the numerator is not smaller than the degree of the denominator.
Your final answer will have an arctangent in it.
1.10.4.15.
Hint.
In the partial fraction decomposition, several constants turn out to be 0.
1.10.4.16.
Hint.
Factor \((2x-1)\) out of the denominator to get started. You don’t need long division for this step.
1.10.4.17.
Hint.
When it comes time to integrate, look for a convenient substitution.
1.10.4.18.
Hint.
\(\displaystyle\csc x = \frac{1}{\sin x} = \frac{\sin x}{\sin^2 x}\)
1.10.4.19.
Hint.
Use the partial fraction decomposition from Queston 18 to save yourself some time.
1.10.4.20.
Hint.
In the final integration, complete the square to make a piece of the integrand look more like the derivative of arctangent.
1.10.4.21.
1.10.4.22.
Hint.
Partial fraction decomposition won’t simplify this any more. Use a trig substitution.
1.10.4.23.
Hint.
To evaluate the antiderivative, break one of the fractions into two fractions.
1.10.4.24.
Hint.
\(\cos^2\theta = 1-\sin^2\theta\)
1.10.4.25.
Hint.
If you’re having a hard time making the substitution, multiply the numerator and the denominator by \(e^x\text{.}\)
1.10.4.26.
Hint.
Try the substitution \(u=\sqrt{1+e^x}\text{.}\) You’ll need to do long division before you can use partial fraction decomposition.
1.10.4.27. (✳).
Hint.
The mechanically easiest way to answer part (c) uses the method of cylindrical shells, which we have not covered. The method of washers also works, but requires you have enough patience and also to have a good idea what \(R\) looks like. So look at the sketch in part (a) very carefully when identifying the left endpoints of your horizontal strips.
1.10.4.28.
Hint.
You’ll need to use two regions, because the curves cross.
1.10.4.29.
Hint.
For (b), use the Fundamental Theorem of Calculus Part 1.
1.11 Numerical Integration
1.11.6 Exercises
1.11.6.1.
Hint.
The absolute error is the difference of the two values; the relative error is the absolute error divided by the exact value; the percent error is one hundred times the relative error.
1.11.6.2.
Hint.
You should have four rectangles in one drawing, and four trapezoids in another.
1.11.6.3.
Hint.
Sketch the second derivative--it’s quadratic.
1.11.6.4.
Hint.
You don’t have to find the actual, exact maximum the second derivative achieves--you only have to give a reasonable “ceiling” that it never breaks through.
1.11.6.5.
Hint.
To compute the upper bound on the error, find an upper bound on the fourth derivative of cosine, then use Theorem 1.11.13 in the text.
To find the actual error, you need to find the actual value of \(A\text{.}\)
1.11.6.6.
Hint.
Find a function with \(f''(x)=3\) for all \(x\) in \([0,1]\text{.}\)
1.11.6.7.
Hint.
You’re allowed to use common sense for this one.
1.11.6.8.
Hint.
For part (b), consider Question 7.
1.11.6.9. (✳).
Hint.
Draw a sketch.
1.11.6.10.
Hint.
The error bound for the approximation is given in Theorem 1.11.13 in the text. You want this bound to be zero.
1.11.6.11.
1.11.6.12. (✳).
Hint.
See Section 1.11.1. You should be able to simplify your answer to an exact value (in terms of \(\pi\)).
1.11.6.13. (✳).
1.11.6.14. (✳).
1.11.6.15. (✳).
1.11.6.18. (✳).
Hint.
The main step is to find an appropriate value of \(M\text{.}\) It is not necessary to find the smallest possible \(M\text{.}\)
1.11.6.19. (✳).
Hint.
The main step is to find \(M\text{.}\) This question is unusual in that its wording requires you to find the smallest possible allowed \(M\text{.}\)
1.11.6.20. (✳).
Hint.
The main steps in part (b) are to find the smallest possible values of \(M\) and \(L\text{.}\)
1.11.6.21. (✳).
Hint.
As usual, the biggest part of this problem is finding \(L\text{.}\) Don’t be thrown off by the error bound being given slightly differently from Theorem 1.11.13 in the text: these expressions are equivalent, since \(\De x = \frac{b-a}{n}\text{.}\)
1.11.6.22. (✳).
Hint.
The function \(e^{-2x} = \dfrac{1}{e^{2x}}\) is positive and decreasing, so its maximum occurs when \(x\) is as small as possible.
1.11.6.23. (✳).
Hint.
Since \(\dfrac{1}{x^5}\) is a decreasing function when \(x \gt 0\text{,}\) look for its maximum value when \(x\) is as small as possible.
1.11.6.24. (✳).
Hint.
The “best ... approximations that you can” means using the maximum number of intervals, given the information available.
The final sentence in part (b) is just a re-statement of the error bounds we’re familiar with from Theorem 1.11.13 in the text. The information \(\big|s^{(k)}(x)\big|\le \dfrac{k}{1000}\) gives you values of \(M\) and \(L\) when you set \(k=2\) and \(k=4\text{,}\) respectively.
1.11.6.25. (✳).
Hint.
Set the error bound to be less than \(0.001\text{,}\) then solve for \(n\text{.}\)
1.11.6.26. (✳).
Hint.
Since the cross-sections of the pool are semi-circular disks, a section that is \(d\) metres across will have area \(\frac{1}{2}\pi\left(\frac{d}{2}\right)^2\) square feet. Based on the drawing, you may assume the very ends of the pool have distance 0 feet across.
1.11.6.27. (✳).
Hint.
See Example 1.11.15.
Don’t get caught up in the interpretation of the integral. It’s nice to see how integrals can be used, but for this problem, you’re still just approximating the integral given, and bounding the error.
When you find the second derivative to bound your error, pay attention to the difference between the integrand and \(g(r)\text{.}\)
1.11.6.28. (✳).
Hint.
See Example 1.11.16. You’ll want to use a calculator for the approximation in (a), and for finding the appropriate number of intervals in (b). Remember that Simpson’s rule requires an even number of intervals.
1.11.6.29. (✳).
Hint.
See Example 1.11.16.
Rather than calculating the fourth derivative of the integrand, use the graph to find the largest absolute value it attains over our interval.
1.11.6.30. (✳).
Hint.
See Example 1.11.15.
You’ll have to differentiate \(f(x)\text{.}\) To that end, you may also want to review the fundamental theorem of calculus and, in particular, Example 1.3.5.
You don’t have to find the best possible value for \(M\text{.}\) A reasonable upper bound on \(|f''(x)|\) will do.
To have five decimal places of accuracy, your error must be less than 0.000005. This ensures that, if you round your approximation to five decimal places, they will all be correct.
1.11.6.31.
Hint.
To find the maximum value of \(|f''(x)|\text{,}\) check its critical points and endpoints.
1.11.6.32.
Hint.
In using Simpson’s rule to approximate \(\displaystyle\int_1^{x\vphantom{\frac{1}{2}}} \frac{1}{t} \dee{t}\) with \(n\) intervals, \(a=1\text{,}\) \(b=x\text{,}\) and \(\De x = \dfrac{x-1}{n}\text{.}\)
1.11.6.33.
Hint.
- \(\int_1^2 \frac{1}{1+x^2} \dee{x} = \arctan(2) - \frac{\pi}{4}\text{,}\) so \(\arctan(2) = \frac{\pi}{4}+\int_1^2 \frac{1}{1+x^2} \dee{x}\)
- If an approximation \(A\) of the integral \(\int_1^2 \frac{1}{1+x^2} \dee{x}\) has error at most \(\varepsilon\text{,}\) then \(A-\varepsilon \leq \int_1^2 \frac{1}{1+x^2} \dee{x} \leq A+\varepsilon\text{.}\)
- Looking at our target interval will tell you how small \(\varepsilon\) needs to be, which in turn will tell you how many intervals you need to use.
- You can show, by considering the numerator and denominator separately, that \(|f^{(4)}(x)| \leq 30.75\) for every \(x\) in \([1,2]\text{.}\)
- If you use Simpson’s rule to approximate \(\int_1^2 \frac{1}{1+x^2} \dee{x}\text{,}\) you won’t need very many intervals to get the requisite accuracy.
1.12 Improper Integrals
1.12.4 Exercises
1.12.4.1.
Hint.
There are two kinds of impropreity in an integral: an infinite discontinuity in the integrand, and an infinite limit of integration.
1.12.4.2.
Hint.
The integrand is continuous for all \(x\text{.}\)
1.12.4.3.
Hint.
What matters is which function is bigger for large values of \(x\text{,}\) not near the origin.
1.12.4.4. (✳).
Hint.
Read both the question and Theorem 1.12.17 very carefully.
1.12.4.5.
Hint.
(a) What if \(h(x)\) is negative? What if it’s not?
(b) What if \(h(x)\) is very close to \(f(x)\) or \(g(x)\text{,}\) rather than right in the middle?
(c) Note \(|h(x)| \leq 2f(x)\text{.}\)
1.12.4.6. (✳).
Hint.
First: is the integrand unbounded, and if so, where?
Second: when evaluating integrals, always check to see if you can use a simple substitution before trying a complicated procedure like partial fractions.
1.12.4.7. (✳).
Hint.
Is the integrand bounded?
1.12.4.8. (✳).
Hint.
See Example 1.12.21. Rather than antidifferentiating, you can find a nice comparison.
1.12.4.9. (✳).
Hint.
Which of the two terms in the denominator is more important when \(x\approx
0\text{?}\) Which one is more important when \(x\) is very large?
1.12.4.10.
Hint.
Remember to break the integral into two pieces.
1.12.4.11.
Hint.
Remember to break the integral into two pieces.
1.12.4.12.
Hint.
The easiest test in this case is limiting comparison, Theorem 1.12.22.
1.12.4.13.
Hint.
Not all discontinuities cause an integral to be improper--only infinite discontinuities.
1.12.4.14. (✳).
Hint.
Which of the two terms in the denominator is more important when \(x\) is very large?
1.12.4.15. (✳).
Hint.
Which of the two terms in the denominator is more important when \(x\approx
0\text{?}\) Which one is more important when \(x\) is very large?
1.12.4.16. (✳).
Hint.
What are the “problem \(x\)’s” for this integral? Get a simple approximation to the integrand near each.
1.12.4.17.
Hint.
To find the volume of the solid, cut it into horizontal slices, which are thin circular disks.
The true/false statement is equivalent to saying that the improper integral giving the volume of the solid when \(a=0\) diverges to infinity.
1.12.4.18. (✳).
Hint.
Review Example 1.12.8. Remember the antiderivative of \(\frac{1}{x}\) looks very different from the antiderivative of other powers of \(x\text{.}\)
1.12.4.19.
Hint.
Compare to Example 1.12.14 in the text. You can antidifferentiate with a \(u\)-substitution.
1.12.4.20.
Hint.
To evaluate the integral, you can factor the denominator.
Recall \(\displaystyle\lim_{x \to \infty}\arctan x = \frac{\pi}{2}\text{.}\) For the other limits, use logarithm rules, and beware of indeterminate forms.
1.12.4.21.
Hint.
Break up the integral. The absolute values give you a nice even function, so you can replace \(|x-a|\) with \(x-a\) if you’re careful about the limits of integration.
1.12.4.22.
Hint.
Use integration by parts twice to find the antiderivative of \(e^{-x}\sin x\text{,}\) as in Example 1.7.10. Be careful with your signs--it’s easy to make a mistake with all those negatives.
If you’re having a hard time taking the limit at the end, review the Squeeze Theorem (see the CLP-1 text).
1.12.4.23. (✳).
Hint.
What is the limit of the integrand when \(x\rightarrow 0\text{?}\)
1.12.4.24.
Hint.
The only “source of impropriety” is the infinite domain of integration. Don’t be afraid to be a little creative to make a comparison work.
1.12.4.25. (✳).
Hint.
There are two things that contribute to your error: using \(t\) as the upper bound instead of infinity, and using \(n\) intervals for the approximation.
First, find a \(t\) so that the error introduced by approximating \(\int_0^\infty \frac{e^{-x}}{1+x}\dee{x}\) by \(\int_0^t \frac{e^{-x}}{1+x}\dee{x}\) is at most \(\frac{1}{2} 10^{-4}\text{.}\) Then, find your \(n\text{.}\)
1.12.4.26.
1.12.4.27.
Hint.
\(x\) should be a real number
1.13 More Integration Examples
Exercises
1.13.1.
Hint.
Each option in each column should be used exactly once.
1.13.2.
Hint.
The integrand is the product of sines and cosines. See how this was handled with a substitution in Section 1.8.1.
After your substitution, you should have a polynomial expression in \(u\)—but it might take some simplification to get it into a form you can easily integrate.
1.13.3.
Hint.
We notice that the integrand has a quadratic polynomial under the square root. If that polynomial were a perfect square, we could get rid of the square root: try a trig substitution, as in Section 1.9.
The identity \(\sin(2\theta)=2\sin\theta\cos\theta\) might come in handy.
1.13.4.
Hint.
Notice the integral is improper. When you compute the limit, l’Hôpital’s rule might help.
If you’re struggling to think of how to antidifferentiate, try writing \(\dfrac{x-1}{e^x} = (x-1)e^{-x}\text{.}\)
1.13.5.
Hint.
Which method usually works for rational functions (the quotient of two polynomials)?
1.13.6.
Hint.
It would be nice to replace logarithm with its derivative, \(\dfrac{1}{x}\text{.}\)
1.13.7. (✳).
Hint.
The integrand is a rational function, so it is possible to use partial fractions. But there is a much easier way!
1.13.8. (✳).
Hint.
You should prepare your own personal internal list of integration techniques ordered from easiest to hardest. You should have associated to each technique your own personal list of signals that you use to decide when the technique is likely to be useful.
1.13.9. (✳).
Hint.
Despite both containing a trig function, the two integrals are easiest to evaluate using different methods.
1.13.10. (✳).
1.13.11. (✳).
Hint.
Part (a) can be done by inspection — use a little highschool geometry! Part (b) is reminiscent of the antiderivative of logarithm—how did we find that one out? Part (c) is an improper integral.
1.13.12.
Hint.
Use the substitution \(u=\sin\theta\text{.}\)
1.13.13. (✳).
Hint.
For (c), try a little algebra to split the integral into pieces that are easy to antidifferentiate.
1.13.14. (✳).
1.13.15. (✳).
1.13.16. (✳).
Hint.
For part (b), first complete the square in the denominator. You can save some work by first comparing the derivative of the denominator with the numerator. For part (d) use a simple substitution.
1.13.17. (✳).
Hint.
For part (b), complete the square in the denominator. You can save some work by first comparing the derivative of the denominator with the numerator.
1.13.18. (✳).
Hint.
For part (a), the numerator is the derivative of a function that appears in the denominator.
1.13.19.
Hint.
The integral is improper.
1.13.20. (✳).
Hint.
For part (a), can you convert this into a partial fractions integral? For part (b), start by completing the square inside the square root.
1.13.21. (✳).
Hint.
For part (b), the numerator is the derivative of a function that is embedded in the denominator.
1.13.22.
Hint.
Try a substitution.
1.13.23.
Hint.
Note the quadratic function under the square root: you can solve this with trigonometric substitution, as in Section 1.9.
1.13.24.
Hint.
Try a substitution, as in Section 1.8.2.
1.13.25.
Hint.
What’s the usual trick for evaluating a rational function (quotient of polynomials)?
1.13.26.
Hint.
If the denominator were \(x^2+1\text{,}\) the antiderivative would be arctangent.
1.13.27.
Hint.
Simplify first.
1.13.28.
Hint.
\(x^3+1 = (x+1)(x^2-x+1)\)
1.13.29.
Hint.
You have the product of two quite dissimilar functions in the integrand—try integration by parts.
1.13.30.
Hint.
Use the identity \(\cos(2x) = 2\cos^2x -1\text{.}\)
1.13.31.
Hint.
Using logarithm rules can make the integrand simpler.
1.13.32.
Hint.
What is the derivative of the function in the denominator? How could that be useful to you?
1.13.33. (✳).
Hint.
For part (a), the substitution \(u=\log x\) gives an integral that you have seen before.
1.13.34. (✳).
Hint.
For part (a), split the integral in two. One part may be evaluated by interpreting it geometrically, without doing any integration at all. For part (c), multiply both the numerator and denominator by \(e^x\) and then make a substitution.
1.13.35.
Hint.
Let \(u=\sqrt{1-x}\text{.}\)
1.13.36.
Hint.
Use the substitution \(u=e^x\text{.}\)
1.13.37.
Hint.
Use integration by parts. If you choose your parts well, the resulting integration will be very simple.
1.13.38.
Hint.
\(\frac{\sin x}{\cos^2 x} = \tan x \sec x\)
1.13.39.
Hint.
The cases \(n=-1\) and \(n=-2\) are different from all other values of \(n\text{.}\)
1.13.40.
Hint.
\(x^4+1 = (x^2+\sqrt2x+1)(x^2-\sqrt2x+1)\)
2 Applications of Integration
2.1 Work
2.1.2 Exercises
2.1.2.1.
Hint.
Watch your units: \(1 \,\mathrm{J} = 1\, \frac{\mathrm{kg}\cdot\mathrm{m}^2}{\mathrm{sec}^2}\text{,}\) but your mass is not given in kilograms, and your height is not given in metres.
2.1.2.2.
Hint.
The force of the rock on the ground is the product of its mass and the acceleration due to gravity.
2.1.2.3.
Hint.
Adding or subtracting two quantities of the same units doesn’t change the units. For example, if I have one metre of rope, and I tie on two more metres of rope, I have \(1+2=3\) metres of rope — not 3 centimetres of rope, or 3 kilograms of rope.
Multiplying or dividing quantities of some units gives rise to a quantity with the product or quotient of those units. For example, if I buy ten pounds of salmon for \(\$\)50, the price of my salmon is \(\dfrac{50\text{ dollars} }{10\text{ pounds}} = \dfrac{50}{10} \frac{\text{dollars}}{\text{pound}} = 5 \frac{\text{dollars}}{\text{pound}}\text{.}\) (Not 5 pound-dollars, or 5 pounds.)
2.1.2.4.
Hint.
See Question 3.
2.1.2.5.
Hint.
Hooke’s law says that the force required to stretch a spring \(x\) units past its natural length is proportional to \(x\text{;}\) that is, there is some constant \(k\) associated with the individual spring such that the force required to stretch it \(x\) m past its natural length is \(kx\text{.}\)
2.1.2.6.
Hint.
Definition 2.1.1 tells us the work done by the force from \(x=1\) to \(x=b\) is \(W(b) = \int_1^b F(x) \dee{x}\text{,}\) where \(F(x)\) is the force on the object at position \(x\text{.}\) To recover the equation for \(F(x)\text{,}\) use the Fundamental Theorem of Calculus.
2.1.2.7. (✳).
Hint.
Review Definition 2.1.1 for calculating the work done by a force over a distance.
2.1.2.8.
Hint.
For (a), \(\frac{c}{\ell-x}\) is meausured in Newtons, while \(\ell\) and \(x\) are in metres. For (b), notice the similarities and differences between the tube of air and a spring obeying Hooke’s law.
2.1.2.9. (✳).
Hint.
See Example 2.1.2. Be careful about your units.
2.1.2.10. (✳).
Hint.
Be careful about the units.
2.1.2.11. (✳).
Hint.
Suppose that the bucket is a distance \(y\) above the ground. How much work is required to raise it an additional height \(\dee{y}\text{?}\)
2.1.2.12.
Hint.
Since you’re given the area of the cross-section, it doesn’t matter what shape it has. However, the density of water is given in cubic centimetres, while the measurements of the tank are given in metres.
2.1.2.13. (✳).
Hint.
Consider the work done to lift a horizontal plate from 2 m below the ground to a height \(z\text{.}\) You’ll need to know the mass of the plate, which you can calculate from its volume, since its density is given to you.
2.1.2.14.
Hint.
You can find the spring constant \(k\) from the information about the hanging kilogram.
2.1.2.15.
2.1.2.16.
Hint.
Calculating the work done on the rope and the weight separately makes the computation somewhat easier.
2.1.2.17.
Hint.
When you pull the box, the force you’re exerting is exactly the same as the frictional force, but in the opposite direction. In (a), that force is constant. In (b), it changes. Check Definition 2.1.1 for how to turn force into work.
2.1.2.18.
Hint.
Remember that the work done on an object is equal to the change in its kinetic energy, which is \(\frac{1}{2}mv^2\text{,}\) where \(m\) is the mass of the object and \(v\) is its velocity. Hooke’s law will tell you how much work was done stretching the spring.
2.1.2.19.
Hint.
As in Question 18 in this section, the change in kinetic energy of the car is equal to the work done by the compressing struts. The only added step is to calculate the spring constant, given that a car with mass 2000 kg compresses the spring 2 cm in Earth’s gravity. You’re not calculating work to find the spring constant: you’re using the fact that when the car is sitting still, the force exerted upward by the struts is equal to the force exerted downward by the mass of the car under gravity.
2.1.2.20.
Hint.
To find the radius of a horizontal layer of water, use similar triangles. Be careful with centimetres versus metres.
2.1.2.21. (✳).
Hint.
See Example 2.1.4 for a basic method for calculating the work done pumping water.
To find the area of a horizontal layer of water, use some geometry. A horizontal cross-section of a sphere is a circle, and its radius will depend on the height of the layer in the tank.
2.1.2.22.
Hint.
The basic ideas you’ve used already with “cable problems” still work, you only need to take care that the density of the cable is no longer constant. The mass of a tiny piece of cable, say of length \(\dee{x}\text{,}\) is (density)\(\times\)(length) = \((10-x)\dee{x}\text{,}\) where \(x\) is the distance of our piece from the bottom of the cable.
2.1.2.23.
Hint.
To calculate the force on the entire plunger, first find the force on a horizontal rectangle with height \(\dee{y}\) at depth \(y\text{.}\)
Checking units can be a good way to make sure your calculation makes sense.
2.1.2.24.
Hint.
When \(y\) metres of rope have been hauled up, what is the mass of the water?
2.1.2.25.
Hint.
The work you’re asked for is an improper integral, moving the earth and moon infinitely far apart.
2.1.2.26.
Hint.
You can formulate a guess by considering the work done on the ball versus the work done on the rope in Question 16, Solution 1. But be careful — the ball in that problem did not have the same mass as the rope.
2.1.2.27.
Hint.
There are two things that vary with height: the density of the liquid, and the area of the cross-section of the tank. Make a formula \(M(h)\) for the mass of a thin layer of liquid \(h\) metres below the top of the tank, using mass\(=\)volume\(\times\)density. The rest of the problem is similar to other tank-pumping problems in this section.
2.1.2.28.
Hint.
You can model the motion, instead of a rotation, as dividing the sand into thin horizontal slices and lifting each of them to their new position.
- In order to calculate the work involved lifting a layer of sand, you need to know the mass of the layer of sand.
- To find the mass of a layer of sand, you need its volume and the density of the sand.
- To find the density of the sand, you need to the volume of the sand: that is, the volume of half the hourglass.
- The hourglass is a solid of rotation: you can find its volume using an integral, as in Section 1.6.
2.1.2.29.
Hint.
Theorem 1.11.13 gives error bounds for the standard types of numerical approximations. You won’t need very many intervals to achieve the desired accuracy.
2.2 Averages
2.2.2 Exercises
2.2.2.1.
Hint.
See Definition 2.2.2 and the discussion following it for the link between area under the curve and averages.
2.2.2.2.
Hint.
Average velocity is discussed in Example 2.2.5. You don’t need an integral for this.
2.2.2.3.
Hint.
Much like Problem 2, you don’t need to do any integration here.
2.2.2.4.
Hint.
Part (a) is asking the length of the pieces we’ve cut our interval into. Part (c) should be given in terms of \(f\text{.}\) Our final answer in (d) will resemble a Riemann sum, but without some extra manipulation it won’t be in exactly the form of a Riemann sum we’re used to.
2.2.2.5.
Hint.
For (b), the value of \(f(0)\) could be much, much larger than \(g(0)\text{.}\)
2.2.2.6.
Hint.
The answer is something very simple.
2.2.2.7. (✳).
Hint.
Apply the definition of “average value” in Section 2.2.
2.2.2.8. (✳).
Hint.
You can antidifferentiate \(x^2\log x\) using integration by parts.
2.2.2.9. (✳).
Hint.
You can antidifferentiate an odd power of cosine with a substitution; for an even power of cosine, use the identity \(\cos^2 x = \frac{1}{2}\big(1+\cos(2x)\big)\text{.}\)
2.2.2.10. (✳).
Hint.
If you’re not sure how to antidifferentiate, try the substitution \(u=kx\text{,}\) \(\dee{u}=k\dee{x}\text{,}\) keeping in mind that \(k\) is a constant. Interestingly, your final answer won’t depend on \(k\text{.}\)
2.2.2.11. (✳).
Hint.
The method of partial fractions can help you antidifferentiate.
2.2.2.12. (✳).
Hint.
Try the substitution \(u=\log x\text{,}\) \(\dee{u}=\frac{1}{x}\,\dee{x}\text{.}\)
2.2.2.13. (✳).
Hint.
Remember \(\cos^2 x = \frac{1}{2}\big(1+\cos(2x)\big)\text{.}\)
2.2.2.14.
Hint.
Notice the term \(50\cos\left(\frac{t}{12}\pi\right)\) has a period of 24 hours, while the term \(200\cos\left(\frac{t}{4380}\pi\right)\) has a period of one year.
If \(n\) is an approximation of \(c\text{,}\) then the relative error of \(n\) is \(\frac{|n-c|}{c}\text{.}\)
2.2.2.15.
Hint.
A cross section of \(S\) at location \(x\) is a circle with radius \(x^2\text{,}\) so area \(\pi x^4\text{.}\) Part (a) is asking for the average of this function on \([0,2]\text{.}\)
2.2.2.16.
Hint.
(a) can be done without calculation
2.2.2.17.
Hint.
\(\tan^2 x = \sec^2 x - 1\)
2.2.2.18.
Hint.
Remember force is the product of the spring constant with the distance it’s stretched past its natural length. The units given in the question are not exactly standard, but they are compatible with each other.
You can find part (b) without any calculation. For (c), remember \(\sin^2 x = \frac{1}{2}\big(1-\cos(2x)\big)\text{.}\)
2.2.2.19. (✳).
Hint.
The trapezoidal rule is found in Section 1.11.2.
2.2.2.20.
Hint.
To find a definite integral of the absolute value of a function, break up the interval of integration into regions where the function is positive, and intervals where it’s negative.
2.2.2.21.
Hint.
This is an application of the ideas in Question 20.
2.2.2.22.
Hint.
Slice the solid into circular disks of radius \(|f(x)|\) and thickness \(\dee{x}\text{.}\)
2.2.2.23.
Hint.
The question tells you \(\frac{1}{1-0}\int_0^1 f(x)\,\dee{x} = \frac{f(0)+f(1)}{2}\text{.}\)
2.2.2.24.
Hint.
Set up this question just like Question 23, but with variables for your limits of integration.
Note \((s-t)^2=s^2-2st+t^2\text{.}\)
2.2.2.25.
Hint.
What are the graphs of \(f(x)\) and \(f(a+b-x)\) like?
2.2.2.26.
Hint.
For (b), express \(A(x)\) as an integral, then differentiate.
2.2.2.27.
Hint.
For (b), consider the cases that \(f(x)\) is always bigger or always smaller than 0. Then, use the intermediate value theorem (see the CLP-1 text).
2.2.2.28.
Hint.
Try l’Hôpital’s rule.
2.2.2.29.
Hint.
Use the result of Question 28.
2.3 Centre of Mass and Torque
2.3.3 Exercises
2.3.3.1.
Hint.
It might help to know that \(-x^2+2x+1 = 2-(x-1)^2\text{.}\)
2.3.3.2.
Hint.
The centroid of a region doesn’t have to be a point in the region.
2.3.3.3.
2.3.3.4.
Hint.
Use Equation 2.3.1.
2.3.3.5.
Hint.
Imagine cutting out the shape and setting it on top of a pencil, so that the pencil lines up with the vertical line \(x=a\text{.}\) Will the figure balance, or fall to one side? Which side?
2.3.3.6.
Hint.
You can find the heights of the centres of mass using symmetry.
2.3.3.7.
Hint.
Think about whether your answers should have repetition.
2.3.3.8.
Hint.
The definition of a definite integral (Definition 1.1.9) will tell you how to convert your limits of sums into integrals.
2.3.3.9.
Hint.
In (a), the slices all have the same width, so the area of the slices is larger (and hence the density of \(R\) is higher) where \(T(x)-B(x)\) is larger.
2.3.3.10.
Hint.
Part (a) is a significantly different model from the last question.
2.3.3.11. (✳).
Hint.
Which method involves more work: horizontal strips or vertical strips?
2.3.3.12.
Hint.
This is a straightforward application of Equation 2.3.4.
2.3.3.13.
Hint.
Remember the derivative of arctangent is \(\frac{1}{1+x^2}\)
2.3.3.14. (✳).
2.3.3.15. (✳).
Hint.
You can use a trigonometric substitution to find the area, then a partial fraction decomposition to find the \(y\)-coordinate of the centroid. Remember \(\sin(1/2)=\pi/6\text{.}\)
2.3.3.16. (✳).
Hint.
Vertical slices will be easier than horizontal. An integration by parts might be helpful to find \(\bar x\text{,}\) while trigonometric identities are important to finding \(\bar y\text{.}\)
2.3.3.17. (✳).
Hint.
No trigonometric substitution is necessary if you’re clever with your \(u\)-substitutions, and remember the derivative of arctangent.
2.3.3.18. (✳).
Hint.
In \(R\text{,}\) the top function is \(x-x^2\text{,}\) and the bottom function is \(x^2-3x\text{.}\)
2.3.3.19. (✳).
Hint.
Remember \(\diff{}{x}\{\arctan x\} = \frac{1}{1+x^2}\text{.}\)
2.3.3.20. (✳).
Hint.
You can save quite a bit of work by, firstly, exploiting symmetry and, secondly, thinking about whether it is more efficient to use vertical strips or horizontal strips.
2.3.3.21. (✳).
Hint.
Sketch the region, being careful the domain of \(\sqrt{9-4x^2}\text{.}\) You can save quite a bit of work by exploiting symmetry.
2.3.3.22.
Hint.
Horizontal slices will be easier than vertical.
2.3.3.23.
Hint.
Start with a picture: whether you use vertical slices or horizontal, you’ll need to break your integral into multiple pieces.
2.3.3.24. (✳).
Hint.
For practice, do the computation twice — once with horizontal strips and once with vertical strips. Watch for improper integrals.
2.3.3.25. (✳).
Hint.
Draw a sketch. In part (b) be careful about the equation of the right hand boundary of \(A\text{.}\)
2.3.3.26. (✳).
Hint.
Draw a sketch. Rotating about a horizontal line is similar to rotating about the \(x\)-axis, but for the radius of a slice, you’ll need to know \(|y-(-1)|\text{:}\) the distance from the outer edge of the region (the boundary function’s \(y\)-value) to \(y=-1\text{.}\)
2.3.3.27.
Hint.
Go back to the derivation of Equation 2.3.5 (centroid for a region) to figure out what to do when your surface does not have uniform density. We will consider a rod \(R\) that reaches from \(x=0\) to \(x=4\text{,}\) and the mass of the section of the rod along \([a,b]\) is equal to the mass of the strip of our rectangle along \([a,b]\text{.}\)
2.3.3.28.
Hint.
Horizontal slices will help you, where symmetry doesn’t, to set up a rod \(R\) whose centre of mass is the same as one coordinate of the centre of mass of the circle. When you’re integrating, trigonometric substitutions are sometimes the easiest way, and sometimes not.
The equation of a circle of radius 3, centred at \((0,3)\text{,}\) is \(x^2+(y-3)^2=9\text{.}\)
2.3.3.29.
Hint.
The model in the question gives you the setup to solve this problem. You know how to find the centre of mass of a rod — that’s Equation 2.3.4 — so all you need to find is \(\rho(y)\text{,}\) the density of the rod at position \(y\text{.}\) To find this, consider a thin slice of the cone at position \(y\) with thickness \(\dee{y}\text{.}\) Its volume \(V(y)\) is the same as the mass of the small section of the rod at position \(y\) with thickness \(\dee{y}\text{.}\) So, the density of the rod at position \(y\) is \(\rho(y)=\frac{V(y)}{\dee{y}}\text{.}\)
2.3.3.30.
Hint.
Use similar triangles to show that the shape of the lower (also upper) half of the hourglass is a truncated cone, where the untruncated cone would have had a height 10 cm.
2.3.3.31.
Hint.
The techniques of Section 2.1 get pretty complicated here, so it’s easiest to use the techniques we developed in Questions 6, 29, and 30 in this section. That is, (1) find the height of the centre of mass of the water in its starting and ending positions, and then (2) model the work done as the work moving a point mass with the weight of the water from the first centre of mass to the second.
The height change of the centre of mass is all that matters to calculate the work done against gravity, so you only have to worry about the height of the centres of mass.
2.3.3.32.
Hint.
The area of \(R\) is precisely one, so the error in your approximation is the error involved in approximating \(\int_0^{\sqrt{\pi/2}}2x^2\sin(x^2)\,\dee{x}\text{.}\)
2.4 Separable Differential Equations
2.4.7 Exercises
2.4.7.1.
Hint.
You don’t need to solve the differential equation from scratch, only verify whether the given function \(y=f(x)\) makes it true. Find \(\diff{y}{x}\) and plug it into the differential equation.
2.4.7.2.
Hint.
For (d), note the equation given is quadratic in the variable \(\diff{y}{x}\text{.}\)
2.4.7.3.
Hint.
The step \(\displaystyle\int \frac{1}{g(y)}\,\dee{y} = \int f(x)\,\dee{x}\) shows up whether we’re using our mnemonic or not.
2.4.7.4.
Hint.
Note \(\diff{}{x}\{f(x)\} = \diff{}{x}\{f(x)+C\}\text{.}\) Plug in \(y=f(x)+C\) to the equation \(\diff{y}{x}=xy\) to see whether it makes the equation is true.
2.4.7.5.
Hint.
If a function is differentiable at a point, it is also continuous at that point.
2.4.7.6.
Hint.
Let \(Q(t)\) be the quantity of morphine in a patient’s bloodstream at time \(t\text{,}\) where \(t\) is measured in minutes.
Using the definition of a derivative,
\begin{equation*}
\diff{Q}{t}=\lim_{h \to 0}\frac{Q(t+h)-Q(t)}{h}\approx \frac{Q(t+1)-Q(t)}{1}
\end{equation*}
So, \(\diff{Q}{t}\) is roughly the change in the amount of morphine in one minute, from \(t\) to \(t+1\text{.}\)
2.4.7.7.
Hint.
If \(p(t)\) is the proportion of the new form, then \(1-p(t)\) is the proportion of the old form.
When we say two quantities are proportional, we mean that one is a constant multiple of the other.
2.4.7.8.
Hint.
The red marks show the slope \(y(x)\) would have at a point if it crosses that point. So, pick a value of \(y(0)\text{;}\) based on the red marks, you can see how fast \(y(x)\) is increasing or decreasing at that point, which leads you roughly to a value of \(y(1)\text{;}\) again, the red marks tell you how fast \(y(x)\) is increasing or decreasing, which leads you to a value of \(y(2)\text{,}\) etc (unless you’re already off the graph).
2.4.7.9.
2.4.7.10. (✳).
Hint.
Start by multiplying both sides of the equation by \(e^y\) and \(\dee{x}\text{,}\) pretending that \(\diff{y}{x}\) is a fraction, according to our mnemonic.
2.4.7.11. (✳).
Hint.
You need to solve for your function \(y(x)\) explicitly. Be careful with absolute values: if \(|y|=F\text{,}\) then \(y=F\) or \(y=-F\text{.}\) However, \(y=\pm F\) is not a function. You have to choose one: \(y=F\) or \(y=-F\text{.}\)
2.4.7.12. (✳).
Hint.
If your answer doesn’t quite look like the answer given, try manipulating it with logarithm rules: \(\log a + \log b = \log(ab)\text{,}\) and \(a\log b = \log(b^a)\text{.}\)
2.4.7.13. (✳).
Hint.
Simplify the equation.
2.4.7.14. (✳).
Hint.
Be careful with the arbitrary constant.
2.4.7.15. (✳).
Hint.
Start by cross-multiplying.
2.4.7.16. (✳).
Hint.
Be careful about signs. If \(y^2=F\text{,}\) then possibly \(y=\sqrt{F}\text{,}\) and possibly \(y=-\sqrt{F}\text{.}\) However, \(y=\pm\sqrt{F}\) is not a function.
2.4.7.17. (✳).
Hint.
Be careful about signs.
2.4.7.18. (✳).
Hint.
Be careful about signs. If \(\log|y|=F\text{,}\) then \(|y|=e^F\text{.}\) Since you should give your answer as an explicit function \(y(x)\text{,}\) you need to decide whether \(y=e^F\) or \(y=-e^F\text{.}\)
2.4.7.19. (✳).
Hint.
Move the \(y\) from the left hand side to the right hand side, then use partial fractions to integrate.
Be careful about the signs. Remember that we need \(y=-1\) when \(x=1\text{.}\) This suggests how to deal with absolute values.
2.4.7.20. (✳).
Hint.
The unknown function \(f(x)\) satisfies an equation that involves the derivative of \(f\text{.}\)
2.4.7.21. (✳).
Hint.
Try guessing the partial fractions expansion of \(\dfrac{1}{x(x+1)}\text{.}\)
Since \(x=1\) is in the domain and \(x=0\) is not, you may assume \(x \gt 0\) for all \(x\) in the domain.
2.4.7.22. (✳).
Hint.
\(\displaystyle\diff{}{x}\{\sec x\} = \sec x \tan x\)
2.4.7.23. (✳).
Hint.
The general solution to the differential equation will contain the constant \(k\) and one other constant. They are determined by the data given in the question.
2.4.7.24. (✳).
Hint.
- When you’re solving the differential equation, you should have an integral that you can massage to look something like arctangent.
- What is the velocity of the object at its highest point?
- Your final answer will depend on the (unspecified) constants \(v_0\text{,}\) \(m\text{,}\) \(g\) and \(k\text{.}\)
2.4.7.25. (✳).
Hint.
The general solution to the differential equation will contain the constant \(k\) and one other constant. They are determined by the data given in the question.
2.4.7.26. (✳).
Hint.
The method of partial fractions will help you integrate.
To solve \(\frac{x-a}{x-b}=Y\) for \(x\text{,}\) move the terms containing \(x\) out of the denominator, then gather them on one side of the equals sign and factor out the \(x\text{.}\)
\begin{align*}
\frac{\textcolor{red}{x}-a}{\textcolor{red}{x}-b}&=Y\\
\textcolor{red}{x}-a&=Y(\textcolor{red}{x}-b)=Y\textcolor{red}{x}-Yb\\
\textcolor{red}{x}-Y\textcolor{red}{x}&=a-Yb\\
\textcolor{red}{x}(1-Y)&=a-Yb\\
\textcolor{red}{x}=\frac{a-Yb}{1-Y}
\end{align*}
To find the limit, you can avoid l’Hôpital’s rule using some clever algebra--but you can also just use l’Hôpital’s rule.
2.4.7.27. (✳).
Hint.
Be careful about signs.
Part (a) has some algebraic similarities to Question 26.
2.4.7.28. (✳).
Hint.
The general solution to the differential equation will contain a constant of proportionality and one other constant. They are determined by the data given in the question.
2.4.7.29. (✳).
Hint.
You do not need to know anything about investing or continuous compounding to do this problem. You are given the differential equation explicitly. The whole first sentence is just window dressing.
2.4.7.30. (✳).
Hint.
Again, you do not need to know anything about investing to do this problem. You are given the differential equation explicitly.
2.4.7.31. (✳).
Hint.
Differentiate the given integral equation. Plugging in \(x=0\) gives you \(y(0)\text{.}\)
2.4.7.32. (✳).
Hint.
Suppose that in a very short time interval \(\dee{t}\text{,}\) the height of water in the tank changes by \(\dee{h}\) (which is negative). Express in two different ways the volume of water that has escaped during this time interval. Equating the two gives the needed differential equation.
As the water escapes, it forms a cylinder of radius 1 cm.
2.4.7.33. (✳).
Hint.
Sketch the mercury in the tank at time \(t\text{,}\) when it has height \(h\text{,}\) and also at time \(t+\dee{t}\text{,}\) when it has height \(h+\dee{h}\) (with \(\dee{h} \lt 0\)). The difference between those two volumes is the volume of (essentially) a disk of thickness \(-\dee{h}\text{.}\) Figure out the radius and then the volume of that disk. This volume has to be the same as the volume of mercury that left through the hole in the bottom of the sphere, which runs out in the shape of a cylinder. Toricelli’s law tells you what the length of that cylinder is, and from there you can find its volume. Setting the two volumes equal to each other gives the differential equation that determines \(h(t)\text{.}\)
2.4.7.34. (✳).
Hint.
The fundamental theorem of calculus will be useful in part (b).
2.4.7.35. (✳).
Hint.
For any \(p \gt 0\text{,}\) determine first \(y(t)\) (in terms of \(p\) and \(c\)) and then the times (also depending on \(p\) and \(c\)) at which \(y=2\text{,}\) \(y=1\) and \(y=0\text{.}\) The condition that “the top half takes exactly the same amount of time to drain as the bottom half” then gives an equation that determines \(p\text{.}\)
2.4.7.36.
Hint.
For (a), think of a very simple function.
The equation in the question statement is equivalent to the equation
\begin{equation*}
\frac{1}{\sqrt{x-a}}\int_a^xf(t)\,\dee(t)=\sqrt{\int_a^x f^2(t)\,\dee{t}}
\end{equation*}
which is, in some cases, easier to use.
For (d), you’ll want to let \(Y(x)=\int_a^x f(t)\,\dee{t}\text{,}\) and use the quadratic equation.
2.4.7.37.
Hint.
Start by antidifferentiating both sides of the equation with respect to \(x\text{.}\)
3 Sequences and series
3.1 Sequences
3.1.2 Exercises
3.1.2.1.
Hint.
Not every limit exists.
3.1.2.2.
Hint.
100 isn’t all that big when you’re contemplating infinity. (Neither is any other number.)
3.1.2.3.
Hint.
\(\displaystyle\lim_{n \to \infty}a_{2n+5}=\lim_{n \to \infty}a_{n}\)
3.1.2.4.
Hint.
The sequence might be defined by different functions when \(n\) is large than when \(n\) is small.
3.1.2.5.
Hint.
Recall \((-1)^n\) is positive when \(n\) is even, and negative when \(n\) is odd.
3.1.2.6.
Hint.
Modify your answer from Question 5, but make the terms approach zero.
3.1.2.7.
Hint.
\((-n)^{-n} = \dfrac{(-1)^n}{n^n}\)
3.1.2.8.
Hint.
What might cause your answers in (a) and (b) to differ? Carefully read Theorem 3.1.6 about convergent functions and their corresponding sequences.
3.1.2.9.
Hint.
You can use the fact that \(\pi\) is somewhat close to \(\dfrac{22}{7}\text{,}\) or you can use trial and error.
3.1.2.10.
Hint.
You can compare the leading terms, or factor a high power of \(n\) from the numerator and denominator.
3.1.2.11.
Hint.
This isn’t a rational expression, but you can treat it in a similar way. Recall \(e \lt 3\text{.}\)
3.1.2.12.
Hint.
The techniques of evaluating limits of rational sequences are again useful here.
3.1.2.13.
Hint.
Use the squeeze theorem.
3.1.2.14.
Hint.
\(\displaystyle\frac{1}{n}\leq n^{\sin n}\leq n\)
3.1.2.15.
Hint.
\(e^{-1/n} = \dfrac{1}{e^{1/n}}\text{;}\) what happens to \(\dfrac{1}{n}\) as \(n\) grows?
3.1.2.16.
Hint.
Use the squeeze theorem.
3.1.2.17.
Hint.
L’Hôpital’s rule might help you decide what happens if you are unsure.
3.1.2.18. (✳).
Hint.
Simplify \(a_k\text{.}\)
3.1.2.19. (✳).
Hint.
What happens to \(\dfrac{1}{n}\) as \(n\) gets very big?
3.1.2.20. (✳).
Hint.
\(\cos 0 =1\)
3.1.2.21. (✳).
Hint.
This is trickier than it looks. Write \(\dfrac{1}{n}=x\) and look at the limit as \(x\rightarrow 0\text{.}\)
3.1.2.22.
Hint.
Multiply and divide by the conjugate.
3.1.2.23.
Hint.
Compared to Question 22, there’s an easier path.
3.1.2.24.
Hint.
Consider \(f'(x)\text{,}\) when \(f(x)=x^{100}\text{.}\)
3.1.2.25.
Hint.
Look to Question 24 for inspiration.
3.1.2.26.
Hint.
The area of an isosceles triangle with two sides of length 1, meeting at an angle \(\theta\text{,}\) is \(\frac{1}{2}\sin\theta\text{.}\)
3.1.2.27.
Hint.
Every term of \(A_n\) is the same, and \(g(x)\) is a constant function.
3.1.2.28.
Hint.
You’ll need to use a logarithm before you can apply l’Hôpital’s rule.
3.1.2.29.
Hint.
(a) Write out the first few terms of the sequence.
(c) Consider how \(a_{n+1}-L\) relates to \(a_{n}-L\text{.}\) What should happen to these numbers if \(a_n\) converges to \(L\text{?}\)
3.1.2.30.
Hint.
Your answer from (b) will help you a lot with the subsequent parts.
3.2 Series
3.2.2 Exercises
3.2.2.1.
Hint.
\(S_N\) is the sum of the terms corresponding to \(n=1\) through \(n=N\text{.}\)
3.2.2.2.
Hint.
Note \(C_k\) is the cumulative number of cookies.
3.2.2.3.
Hint.
How is (a) related to Question 2?
3.2.2.4.
Hint.
You’ll have to calculate \(a_1\) separately from the other terms.
3.2.2.5.
Hint.
When does adding a number decrease the total sum?
3.2.2.6.
Hint.
For (b), imagine cutting up the triangle into its black and white parts, then sharing it equally among a certain number of friends. What is the easiest number of friends to share with, making sure each has the same area in their pile?
3.2.2.7.
Hint.
Compare to Question 6.
3.2.2.8.
Hint.
Iteratively divide a shape into thirds.
3.2.2.9.
Hint.
Lemma 3.2.5 tells us \(\displaystyle\sum_{n=0}^N ar^n = a\dfrac{1-r^{N+1}}{1-r}\text{,}\) for \(r \neq 1\text{.}\)
3.2.2.10.
Hint.
Note \(C_k\) is the cumulative number of cookies.
3.2.2.11.
Hint.
To adjust the starting index, either factor out the first term in the series, or subtract two series. For the subtraction option, consider Question 10.
3.2.2.12.
Hint.
Express your gains in (a) and (c) as series.
3.2.2.13.
Hint.
To find the difference between \(\displaystyle\sum_{n=1}^\infty c_n\) and \(\displaystyle\sum_{n=1}^\infty c_{n+1}\text{,}\) try writing out the first few terms.
3.2.2.14.
Hint.
You might want to first consider a simpler true or false: \(\displaystyle\sum_{n=1}^\infty \dfrac{ a_n}{b_n} \stackrel{?}{=} \dfrac{A}{B}\).
3.2.2.15. (✳).
Hint.
What kind of a series is this?
3.2.2.16. (✳).
Hint.
This is a special kind of series, that you should recognize.
3.2.2.17. (✳).
Hint.
When you see \(\displaystyle\sum_k \Big(\cdots \textcolor{red}{k}\cdots \ \ -\ \ \cdots \textcolor{red}{k+1}\cdots\Big)\text{,}\) you should think “telescoping series.”
3.2.2.18. (✳).
Hint.
When you see \(\displaystyle\sum_n \Big(\cdots \textcolor{red}{n}\cdots \ \ -\ \ \cdots \textcolor{red}{n+1}\cdots\Big)\text{,}\) you should immediately think “telescoping series”. But be careful not to jump to conclusions — evaluate the \(n^{\rm th}\) partial sum explicitly.
3.2.2.19. (✳).
Hint.
Review Definition 3.2.3.
3.2.2.20. (✳).
Hint.
This is a special case of a general series whose sum we know.
3.2.2.21. (✳).
Hint.
Review Example 3.2.6. To write the number as a geometric series, the first few terms might not fit the pattern of the rest of the terms.
3.2.2.22. (✳).
Hint.
Start by writing it as a geometric series.
3.2.2.23. (✳).
Hint.
Review Example 3.2.6. Since the pattern repeats every three decimals, your common ratio \(r\) will be \(\dfrac{1}{10^3}\text{.}\)
3.2.2.24. (✳).
Hint.
Split the series into two parts.
3.2.2.25. (✳).
Hint.
Split the series into two parts.
3.2.2.26. (✳).
Hint.
Split the series into two parts.
3.2.2.27.
Hint.
Use logarithm rules to turn this into a more obvious telescoping series.
3.2.2.28.
Hint.
This is a telescoping series.
3.2.2.29.
Hint.
The stone at position \(x\) has mass \(\dfrac{1}{4^x}\) kg, and we have to pull it a distance of \(2^x\) metres. From this, you can find the work involved in pulling up a single stone. Then, add up the work involved in pulling up all the stones.
3.2.2.30.
Hint.
The volume of a sphere of radius \(r\) is \(\dfrac{4}{3}\pi r^3\text{.}\)
3.2.2.31.
Hint.
Use the properties of a telescoping series to simplify the terms.
Recall \(\sin^2\theta+\cos^2\theta=1\text{.}\)
3.2.2.32.
Hint.
Review Question 3 for using the sequence of partial sums.
3.2.2.33.
Hint.
What is the ratio of areas between the outermost (red) ring and the next (blue) ring?
3.3 Convergence Tests
3.3.11 Exercises
3.3.11.1.
Hint.
That is, which series have terms whose limit is not zero?
3.3.11.2.
Hint.
That is, if \(f(x)\) is a function with \(f(n)=a_n\) for all whole numbers \(n\text{,}\) is \(f(x)\) nonnegative and decreasing?
3.3.11.3.
Hint.
This isn’t a trick. It’s meant to give you intuition to the direct comparison test.
3.3.11.4.
3.3.11.5.
Hint.
Think about Question 4 to remind yourself which way the inequalities have to go for direct comparison.
Note that all the comparison series have positive terms, so we don’t need to worry about that part of the limit comparison test.
3.3.11.6.
Hint.
The divergence test is Theorem 3.3.1.
3.3.11.7.
Hint.
The limit is calculated correctly.
3.3.11.8.
Hint.
It is true that \(f(x)\) is positive. What else has to be true of \(f(x)\) for the integral test to apply?
3.3.11.9.
Hint.
Refer to Question 4.
3.3.11.10.
Hint.
The definition of an alternating series is given in the start of Section 3.3.4.
3.3.11.11.
Hint.
For the ratio test to be inconclusive, \(\lim\limits_{n \to \infty}\left|\dfrac{a_{n+1}}{a_n}\right|\) should be 1 or nonexistent.
3.3.11.12.
Hint.
By the divergence test, for a series \(\sum a_n\) to converge, we need \(\lim\limits_{n \to \infty} a_n=0\text{.}\) That is, the magnitude (absolute value) of the terms needs to be getting smaller.
3.3.11.13.
Hint.
If \(f(x)\) is positive and decreasing, then the integral test tells you that the integral and the series either both increase or both decrease. So, in order to find an example with the properties required in the question, you need \(f(x)\) to not be both positive and decreasing.
3.3.11.14. (✳).
3.3.11.15. (✳).
Hint.
Don’t jump to conclusions about properties of the \(a_n\)’s.
3.3.11.16. (✳).
Hint.
Always try the divergence test first (in your head).
3.3.11.17. (✳).
Hint.
Which test should you always try first (in your head)?
3.3.11.18. (✳).
Hint.
Review the integral test, which is Theorem 3.3.5.
3.3.11.19.
Hint.
A comparison might be helpful — try some algebraic manipulation to find a likely series to compare it to.
3.3.11.20.
Hint.
This is a geometric series.
3.3.11.21.
Hint.
Notice that the series is geometric, but it doesn’t start at \(n=0\text{.}\)
3.3.11.22.
Hint.
Note \(n\) only takes integer values: what’s \(\sin(\pi n)\) when \(n\) is an integer?
3.3.11.23.
Hint.
Note \(n\) only takes integer values: what’s \(\cos(\pi n)\) when \(n\) is an integer?
3.3.11.24.
Hint.
What’s the test that you should always think of when you see a factorial?
3.3.11.25.
Hint.
This is a geometric series, but you’ll need to do a little algebra to figure out \(r\text{.}\)
3.3.11.26.
Hint.
Which test fits most often with factorials?
3.3.11.27.
Hint.
Try finding a nice comparison.
3.3.11.28. (✳).
Hint.
With the substitution \(u=\log x\text{,}\) the function \(\dfrac{1}{x(\log x)^{3/2}}\) is easily integrable.
3.3.11.29. (✳).
Hint.
Combine the integral test with the results about \(p\)-series, Example 3.3.6.
3.3.11.30. (✳).
Hint.
Try the substitution \(u=\sqrt{x}\text{.}\)
3.3.11.31. (✳).
3.3.11.32. (✳).
Hint.
What does the summand look like when \(k\) is very large?
3.3.11.33. (✳).
Hint.
What does the summand look like when \(n\) is very large?
3.3.11.34. (✳).
Hint.
\(\cos(n\pi)\) is a sneaky way to write \((-1)^n\text{.}\)
3.3.11.35. (✳).
Hint.
What is the behaviour for large \(k\text{?}\)
3.3.11.36. (✳).
Hint.
When \(m\) is large, \(3m+\sin\sqrt{m}\approx 3m\text{.}\)
3.3.11.37.
Hint.
This is a geometric series, but it doesn’t start at \(n=0\text{.}\)
3.3.11.38. (✳).
Hint.
The series is geometric.
3.3.11.39. (✳).
Hint.
The first series can be written as \(\displaystyle\sum_{n=1}^\infty \frac{1}{2n-1}\) .
3.3.11.41. (✳).
Hint.
What does the summand look like when \(n\) is very large?
3.3.11.42. (✳).
Hint.
Review the alternating series test, which is given in Theorem 3.3.14.
3.3.11.43. (✳).
Hint.
Review the alternating series test, which is given in Theorem 3.3.14.
3.3.11.44. (✳).
Hint.
Review the alternating series test, which is given in Theorem 3.3.14.
3.3.11.46. (✳).
3.3.11.47. (✳).
Hint.
The truncation error arising from the approximation \(\displaystyle\sum_{n=1}^\infty \frac{e^{-\sqrt{n}}}{\sqrt n} \approx \sum_{n=1}^N \frac{e^{-\sqrt{n}}}{\sqrt n}\) is precisely \(E_N = \displaystyle\sum_{n=N+1}^\infty \frac{e^{-\sqrt{n}}}{\sqrt n}\text{.}\) You’ll want to find a bound on this sum using the integral test.
A key observation is that, since \(f(x) = \dfrac{e^{-\sqrt{x}}}{\sqrt{x}}\) is decreasing, we can show that
\begin{equation*}
\frac{e^{-\sqrt{n}}}{\sqrt{n}} \leq \int_{n-1}^n \frac{e^{-\sqrt{x}}}{\sqrt{x}}\ \dee{x}
\end{equation*}
for every \(n \ge 1\text{.}\)
3.3.11.48. (✳).
Hint.
What does the fact that the series \(\sum\limits_{n=0}^{\infty}a_n\) converges guarantee about the behavior of \(a_n\) for large \(n\text{?}\)
3.3.11.49. (✳).
Hint.
What does the fact that the series \(\sum\limits_{n=0}^{\infty}(1-a_n)\) converges guarantee about the behavior of \(a_n\) for large \(n\text{?}\)
3.3.11.50. (✳).
Hint.
What does the fact that the series \(\displaystyle\sum_{n=1}^\infty\frac{na_n-2n+1}{n+1}\) converges guarantee about the behavior of \(a_n\) for large \(n\text{?}\)
3.3.11.51. (✳).
Hint.
What does the fact that the series \(\sum_{n=1}^\infty a_n\) converges guarantee about the behavior of \(a_n\) for large \(n\text{?}\) When is \(x^2\le x\text{?}\)
3.3.11.52.
Hint.
If we add together the frequencies of all the words, they should amount to 100%. We can approximate this sum using ideas from Example 3.3.4.
3.3.11.53.
Hint.
We are approximating a finite sum — not an infinite series. To get greater accuracy, use exact values for the first several terms in the sum, and use an integral to approximate the rest.
3.4 Absolute and Conditional Convergence
3.4.3 Exercises
3.4.3.1. (✳).
Hint.
What is conditional convergence?
3.4.3.2.
Hint.
If \(\sum |a_n|\) converges, then \(\sum a_n\) is guaranteed to converge as well.
(That’s Theorem 3.4.2.) So, one of the blank spaces describes an impossible sequence.
3.4.3.4. (✳).
Hint.
Be careful about the signs.
3.4.3.5. (✳).
Hint.
Does the alternating series test really apply?
3.4.3.6. (✳).
Hint.
What does the summand look like when \(n\) is very large?
3.4.3.7. (✳).
Hint.
What does the summand look like when \(n\) is very large?
3.4.3.8. (✳).
Hint.
This is a trick question. Be sure to verify all of the hypotheses of any convergence test you apply.
3.4.3.9. (✳).
Hint.
Try the substitution \(u=\log x\text{.}\)
3.4.3.10.
Hint.
Show that it converges absolutely.
3.4.3.11.
Hint.
Use a similar method to Queston 10.
3.4.3.12.
Hint.
Show it converges absolutely using a direct comparison test.
3.4.3.13. (✳).
Hint.
For part (a), replace \(n\) by \(x\) in the absolute value of the summand. Can you integrate the resulting function?
3.4.3.14.
Hint.
You don’t need to add up very many terms for this level of accuracy.
3.4.3.15.
Hint.
Use the direct comparison test to show that the series converges absolutely.
3.5 Power Series
3.5.3 Exercises
3.5.3.1.
Hint.
\(f(1)\) is the sum of a geometric series.
3.5.3.2.
Hint.
Calculate \(\displaystyle\diff{}{x}\left\{\frac{(x-5)^n}{n!+2}\right\}\) when \(n\) is a constant.
3.5.3.3.
Hint.
There is only one.
3.5.3.4.
Hint.
Use Theorem 3.5.9.
3.5.3.5. (✳).
Hint.
Review the discussion immediately following Definition 3.5.1.
3.5.3.6. (✳).
Hint.
Review the discussion immediately following Definition 3.5.1.
3.5.3.7. (✳).
Hint.
Review the discussion immediately following Definition 3.5.1.
3.5.3.8. (✳).
Hint.
See Example 3.5.11.
3.5.3.9. (✳).
Hint.
See Example 3.5.11.
3.5.3.15. (✳).
Hint.
Start part (b) by computing the partial sums of \(\ {\displaystyle \sum_{k=1}^\infty \Big(\frac{a_k}{a_{k+1}}
-\frac{a_{k+1}}{a_{k+2}}\Big)}\)
3.5.3.16. (✳).
Hint.
You should know a power series representation for \(\dfrac{1}{1-x}\text{.}\) Use it.
3.5.3.17.
Hint.
You can safely ignore one of the given equations, but not the other.
3.5.3.18. (✳).
Hint.
\(n\ge\log n\) for all \(n\ge 1\text{.}\)
3.5.3.19. (✳).
3.5.3.20. (✳).
Hint.
You know the geometric series expansion of \(\frac{1}{1-x}\text{.}\) What (calculus) operation(s) can you apply to that geometric series to convert it into the given series?
3.5.3.21. (✳).
Hint.
First show that the fact that the series \(\sum_{n=0}^{\infty}(1-b_n)\) converges guarantees that \(\lim_{n\rightarrow\infty}b_n=1\text{.}\)
3.5.3.22. (✳).
Hint.
What does \(a_n\) look like for large \(n\text{?}\)
3.5.3.23.
Hint.
Equation 2.3.1 tells us the centre of mass of a rod with weights \(\{m_n\}\) at positions \(\{x_n\}\) is \(\displaystyle\bar x =\frac{\sum m_nx_n}{\sum m_n}\) .
3.5.3.24.
Hint.
Use the second derivative test.
3.5.3.25.
Hint.
What function has \(\displaystyle\sum_{n=1}^\infty nx^{n-1}\) as its power series representation?
3.5.3.26.
Hint.
The power series representation in Example 3.5.20 is an alternating series when \(x\) is positive.
3.5.3.27.
Hint.
The power series representation in Example 3.5.21 is an alternating series when \(x\) is nonzero.
3.6 Taylor Series
3.6.8 Exercises
3.6.8.1.
Hint.
Which of the functions are constant, linear, and quadratic?
3.6.8.2.
Hint.
You don’t have to actually calculate the entire series \(T(x)\) to answer the question.
3.6.8.3.
Hint.
If you don’t have these memorized, it’s good to be able to derive them. For instance, \(\log(1+x)\) is the antiderivative of \(\dfrac{1}{1+x}\text{,}\) whose Taylor series can be found by modifying the geometric series \(\sum x^n\text{.}\)
3.6.8.4.
Hint.
See Example 3.6.18.
3.6.8.5.
Hint.
The series will bear some resemblance to the Maclaurin series for \(\log(1+x)\text{.}\)
3.6.8.6.
Hint.
The terms \(f^{(n)}(\pi)\) are going to be similar to the terms \(f^{(n)}(0)\) that we used in the Maclaurin series for sine.
3.6.8.7.
Hint.
The Taylor series will look similar to a geometric series.
3.6.8.8.
Hint.
Your answer will depend on \(a\text{.}\)
3.6.8.9. (✳).
Hint.
You should know the Maclaurin series for \(\dfrac{1}{1-x}\text{.}\) Use it.
3.6.8.10. (✳).
Hint.
You should know the Maclaurin series for \(\dfrac{1}{1-x}\text{.}\) Use it.
3.6.8.11. (✳).
Hint.
You should know the Maclaurin series for \(e^x\text{.}\) Use it.
3.6.8.12. (✳).
Hint.
Review Example 3.5.20.
3.6.8.13. (✳).
Hint.
You should know the Maclaurin series for \(\sin x\text{.}\) Use it.
3.6.8.14. (✳).
Hint.
You should know the Maclaurin series for \(e^x\text{.}\) Use it.
3.6.8.15. (✳).
Hint.
You should know the Maclaurin series for \(\arctan(x)\text{.}\) Use it.
3.6.8.16. (✳).
Hint.
You should know the Maclaurin series for \(\dfrac{1}{1-x}\text{.}\) Use it.
3.6.8.17. (✳).
Hint.
Set \((-1)^n\dfrac{x^{2n+1}}{2n+1}=C\dfrac{(-1)^n}{(2n+1)3^n}\text{,}\) for some constant \(C\text{.}\) What are \(x\) and \(C\text{?}\)
3.6.8.18. (✳).
Hint.
There is an important Taylor series, one of the series in Theorem 3.6.7, that looks a lot like the given series.
3.6.8.19. (✳).
Hint.
There is an important Taylor series, one of the series in Theorem 3.6.7, that looks a lot like the given series.
3.6.8.20. (✳).
Hint.
There is an important Taylor series, one of the series in Theorem 3.6.7, that looks a lot like the given series. Be careful about the limits of summation.
3.6.8.21. (✳).
Hint.
There is an important Taylor series, one of the series in Theorem 3.6.7, that looks a lot like the given series.
3.6.8.22. (✳).
Hint.
Split the series into a sum of two series. There is an important Taylor series, one of the series in Theorem 3.6.7, that looks a lot like each of the two series.
3.6.8.23.
Hint.
Try the ratio test.
3.6.8.24.
Hint.
Write it as the sum of two Taylor series.
3.6.8.25. (✳).
Hint.
Can you think of a way to eliminate the odd terms from \(e^x=\displaystyle \sum_{n=0}^\infty \frac{x^n}{n!}\text{?}\)
3.6.8.26.
Hint.
The series you’re adding up are alternating, so it’s simple to bound the error using a partial sum.
3.6.8.27.
Hint.
The Taylor Series is alternating, so bounding the error in a partial-sum approximation is straightforward.
3.6.8.28.
Hint.
The Taylor Series is not alternating, so use Theorem 3.6.3 to bound the error in a partial-sum approximation.
3.6.8.29.
Hint.
The Taylor Series is not alternating, so use Theorem 3.6.3 to bound the error in a partial-sum approximation.
3.6.8.30.
Hint.
Use Theorem 3.6.3 to bound the error in a partial-sum approximation. This theorem requires you to consider values of \(c\) between \(x\) and \(x=0\text{;}\) since \(x\) could be anything from \(-2\) to \(1\text{,}\) you should think about values of \(c\) between \(-2\) and \(1\text{.}\)
3.6.8.31.
Hint.
Use Theorem 3.6.3 to bound the error in a partial-sum approximation.
To bound the derivative over the appropriate range, remember how to find absolute extrema.
3.6.8.32. (✳).
Hint.
See Example 3.6.23.
3.6.8.33. (✳).
Hint.
See Example 3.6.23.
3.6.8.34.
Hint.
Set \(f(x)=\left(1+x+x^2\right)^{2/x}\text{,}\) and find \(\lim\limits_{x\rightarrow 0}\log\left(f(x)\right)\text{.}\)
3.6.8.35.
Hint.
Use the substitution \(y=\dfrac{1}{x}\text{,}\) and compare to Question 34.
3.6.8.36.
Hint.
Start by differentiating \(\displaystyle\sum_{n=0}^\infty x^n\text{.}\)
3.6.8.37.
Hint.
The series bears a resemblance to the Taylor series for arctangent.
3.6.8.38.
Hint.
For simplification purposes, note \(\displaystyle(1)(3)(5)(7)\cdots(2n-1)=\dfrac{(2n)!}{2^n \,n!}\text{.}\)
3.6.8.39. (✳).
Hint.
You know the Maclaurin series for \(\log(1+y)\text{.}\) Use it! Remember that you are asked for a series expansion in powers of \(x-2\text{.}\) So you want \(y\) to be some constant times \(x-2\text{.}\)
3.6.8.40. (✳).
3.6.8.41. (✳).
Hint.
Look at the signs of successive terms in the series.
3.6.8.42. (✳).
Hint.
The magic word is “series”.
3.6.8.43. (✳).
3.6.8.44. (✳).
3.6.8.45. (✳).
3.6.8.46. (✳).
3.6.8.48. (✳).
Hint.
Use the Maclaurin series for \(e^x\text{.}\)
3.6.8.49. (✳).
Hint.
For part (c), compare two power series term-by-term.
3.6.8.50.
Hint.
For Newton’s method, recall we approximate a root of the function \(g(x)\) in iterations: given an approximation \(x_n\text{,}\) our next approximation is \(x_{n+1}=x_n-\dfrac{g(x_n)}{g'(x_n)}\text{.}\)
To gauge your error, note that from approximation to approximation, the first digits stabilize. Keep refining your approximation until the first two digits stop changing.
3.6.8.51.
Hint.
First, modify your known Maclaurin series for arctangent into a Maclaurin series for \(f(x)\text{.}\) This series is not hard to repeatedly differentiate, so use it to find a power series for \(f^{(10)}(x)\text{.}\)
3.6.8.52.
Hint.
Remember \(e^x\) is never negative for any real number \(x\text{.}\)
3.6.8.53.
Hint.
Since \(f(x)\) is odd, \(f(-x)=-f(x)\) for all \(x\) in its domain. Consider the even-indexed terms and odd-indexed terms of the Taylor series.
