Exercises

Subsection 3.4.11 Exercises

Subsubsection Exercises for § 3.4.1

Exercises — Stage 1

1

The graph below shows three curves. The black curve is $y=f(x)\text{,}$ the red curve is $y=g(x)=1+2\sin(1+x)\text{,}$ and the blue curve is $y=h(x)=0.7\text{.}$ If you want to estimate $f(0)\text{,}$ what might cause you to use $g(0)\text{?}$ What might cause you to use $h(0)\text{?}$

Exercises — Stage 2

In this and following sections, we will ask you to approximate the value of several constants, such as $\log(0.93)\text{.}$ A valid question to consider is why we would ask for approximations of these constants that take lots of time, and are less accurate than what you get from a calculator.

One answer to this question is historical: people were approximating logarithms before they had calculators, and these are some of the ways they did that. Pretend you're on a desert island without any of your usual devices and that you want to make a number of quick and dirty approximate evaluations.

Another reason to make these approximations is technical: how does the calculator get such a good approximation of $\log(0.93)\text{?}$ The techniques you will learn later on in this chapter give very accurate formulas for approximating functions like $\log x$ and $\sin x\text{,}$ which are sometimes used in calculators.

A third reason to make simple approximations of expressions that a calculator could evaluate is to provide a reality check. If you have a ballpark guess for your answer, and your calculator gives you something wildly different, you know to double-check that you typed everything in correctly.

For now, questions like Question 3.4.11.2 through Question 3.4.11.4 are simply for you to practice the fundamental ideas we're learning.

2

Use a constant approximation to estimate the value of $\log(x)$ when $x=0.93\text{.}$ Sketch the curve $y=f(x)$ and your constant approximation.

(Remember that in CLP-1 we use $\log x$ to mean the natural logarithm of $x\text{,}$ $\log_e x\text{.}$)

3

Use a constant approximation to estimate $\arcsin(0.1)\text{.}$

4

Use a constant approximation to estimate $\sqrt{3}\tan(1)\text{.}$

Exercises — Stage 3

5

Use a constant approximation to estimate the value of $10.1^3\text{.}$ Your estimation should be something you can calculate in your head.

Subsubsection Exercises for § 3.4.2

Exercises — Stage 1

1

Suppose $f(x)$ is a function, and we calculated its linear approximation near $x=5$ to be $f(x) \approx 3x-9\text{.}$

What is $f(5)\text{?}$
What is $f'(5)\text{?}$
What is $f(0)\text{?}$

2

The curve $y=f(x)$ is shown below. Sketch the linear approximation of $f(x)$ about $x=2\text{.}$

3

What is the linear approximation of the function $f(x)=2x+5$ about $x=a\text{?}$

Exercises — Stage 2

4

Use a linear approximation to estimate $\log(x)$ when $x=0.93\text{.}$ Sketch the curve $y=f(x)$ and your linear approximation.

(Remember that in CLP-1 we use $\log x$ to mean the natural logarithm of $x\text{,}$ $\log_e x\text{.}$)

5

Use a linear approximation to estimate $\sqrt{5}\text{.}$

6

Use a linear approximation to estimate $\sqrt[5]{30}$

Exercises — Stage 3

7

Use a linear approximation to estimate $10.1^3\text{,}$ then compare your estimation with the actual value.

8

Imagine $f(x)$ is some function, and you want to estimate $f(b)\text{.}$ To do this, you choose a value $a$ and take an approximation (linear or constant) of $f(x)$ about $a\text{.}$ Give an example of a function $f(x)\text{,}$ and values $a$ and $b\text{,}$ where the constant approximation gives a more accurate estimation of $f(b)$ than the linear approximation.

9

The function

\begin{equation*} L(x)=\frac{1}{4}x+\frac{4\pi-\sqrt{27}}{12} \end{equation*}

is the linear approximation of $f(x)=\arctan x$ about what point $x=a\text{?}$

Subsubsection Exercises for § 3.4.3

Exercises — Stage 1

1

The quadratic approximation of a function $f(x)$ about $x=3$ is

\begin{equation*} f(x) \approx -x^2+6x \end{equation*}

What are the values of $f(3)\text{,}$ $f'(3)\text{,}$ $f''(3)\text{,}$ and $f'''(3)\text{?}$

2

Give a quadratic approximation of $f(x)=2x+5$ about $x=a\text{.}$

Exercises — Stage 2

3

Use a quadratic approximation to estimate $\log(0.93)\text{.}$

(Remember that in CLP-1 we use $\log x$ to mean the natural logarithm of $x\text{,}$ $\log_e x\text{.}$)

4

Use a quadratic approximation to estimate $\cos\left(\dfrac{1}{15}\right)\text{.}$

5

Calculate the quadratic approximation of $f(x)=e^{2x}$ about $x=0\text{.}$

6

Use a quadratic approximation to estimate $5^{\tfrac{4}{3}}\text{.}$

7

Evaluate the expressions below.

$\ds\sum_{n=5}^{30} 1$
$\ds\sum_{n=1}^{3} \left[ 2(n+3)-n^2 \right]$
$\ds\sum_{n=1}^{10} \left[\frac{1}{n}-\frac{1}{n+1}\right]$
$\ds\sum_{n=1}^{4}\frac{5\cdot 2^n}{4^{n+1}} $

8

Write the following in sigma notation:

$1+2+3+4+5$
$2+4+6+8$
$3+5+7+9+11$
$9+16+25+36+49$
$9+4+16+5+25+6+36+7+49+8$
$8+15+24+35+48$
$3-6+9-12+15-18$

Exercises — Stage 3

9

Use a quadratic approximation of $f(x)=2\arcsin x$ about $x=0$ to approximate $f(1)\text{.}$ What number are you approximating?

10

Use a quadratic approximation of $e^x$ to estimate $e$ as a decimal.

11

Group the expressions below into collections of equivalent expressions.

$\ds\sum_{n=1}^{10} 2n$
$\ds\sum_{n=1}^{10} 2^n$
$\ds\sum_{n=1}^{10} n^2$
$2\ds\sum_{n=1}^{10} n$
$2\ds\sum_{n=2}^{11} (n-1)$
$\ds\sum_{n=5}^{14} (n-4)^2$
$\dfrac{1}{4}\ds\sum_{n=1}^{10}\left( \frac{4^{n+1}}{2^n}\right)$

Subsubsection Exercises for § 3.4.4

Exercises — Stage 1

1

The 3rd degree Taylor polynomial for a function $f(x)$ about $x=1$ is

\begin{equation*} T_3(x)=x^3-5x^2+9x \end{equation*}

What is $f''(1)\text{?}$

2

The $n$th degree Taylor polynomial for $f(x)$ about $x=5$ is

\begin{equation*} T_n(x)=\sum_{k=0}^{n} \frac{2k+1}{3k-9}(x-5)^k \end{equation*}

What is $f^{(10)}(5)\text{?}$

Exercises — Stage 3

3

The 4th-degree Maclaurin polynomial for $f(x)$ is

\begin{equation*} T_4(x)=x^4-x^3+x^2-x+1 \end{equation*}

What is the third-degree Maclaurin polynomial for $f(x)\text{?}$

4

The 4th degree Taylor polynomial for $f(x)$ about $x=1$ is

\begin{equation*} T_4(x)=x^4+x^3-9 \end{equation*}

What is the third degree Taylor polynomial for $f(x)$ about $x=1\text{?}$

5

For any even number $n\text{,}$ suppose the $n$th degree Taylor polynomial for $f(x)$ about $x=5$ is

\begin{equation*} \sum_{k=0}^{n/2} \frac{2k+1}{3k-9}(x-5)^{2k} \end{equation*}

What is $f^{(10)}(5)\text{?}$

6

The third-degree Taylor polynomial for $f(x)=x^3\left[2\log x - \dfrac{11}{3}\right]$ about $x=a$ is

\begin{equation*} T_3(x)=-\frac{2}{3}\sqrt{e^3}+3ex-6\sqrt{e}x^2+x^3 \end{equation*}

What is $a\text{?}$

Subsubsection Exercises for § 3.4.5

Exercises — Stage 1

1

Give the 16th degree Maclaurin polynomial for $f(x)=\sin x+ \cos x\text{.}$

2

Give the 100th degree Taylor polynomial for $s(t)=4.9t^2-t+10$ about $t=5\text{.}$

3

Write the $n$th-degree Taylor polynomial for $f(x)=2^x$ about $x=1$ in sigma notation.

4

Find the 6th degree Taylor polynomial of $f(x)=x^2\log x+2x^2+5$ about $x=1\text{,}$ remembering that $\log x$ is the natural logarithm of $x\text{,}$ $\log_ex\text{.}$

5

Give the $n$th degree Maclaurin polynomial for $\dfrac{1}{1-x}$ in sigma notation.

Exercises — Stage 3

6

Calculate the $3$rd-degree Taylor Polynomial for $f(x)=x^x$ about $x=1\text{.}$

7

Use a 5th-degree Maclaurin polynomial for $6\arctan x$ to approximate $\pi\text{.}$

8

Write the $100$th-degree Taylor polynomial for $f(x)=x(\log x -1)$ about $x=1$ in sigma notation.

9

Write the $(2n)$th-degree Taylor polynomial for $f(x)=\sin x$ about $x=\dfrac{\pi}{4}$ in sigma notation.

10

Estimate the sum below

\begin{equation*} 1+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}+\cdots +\frac{1}{157!} \end{equation*}

by interpreting it as a Maclaurin polynomial.

11

Estimate the sum below

\begin{equation*} \sum_{k=0}^{100}\frac{(-1)^k}{2k!}\left(\frac{5\pi}{4}\right)^{2k} \end{equation*}

by interpreting it as a Maclaurin polynomial.

Subsubsection Exercises for § 3.4.6

Exercises — Stage 1

1

In the picture below, label the following:

\begin{equation*} f(x) \qquad f\left(x+\Delta x\right) \qquad \Delta x \qquad \Delta y \end{equation*}

2

At this point in the book, every homework problem takes you about 5 minutes. Use the terms you learned in this section to answer the question: if you spend 15 minutes more, how many more homework problems will you finish?

Exercises — Stage 2

3

Let $f(x)=\arctan x\text{.}$

Use a linear approximation to estimate $f(5.1)-f(5)\text{.}$
Use a quadratic approximation to estimate $f(5.1)-f(5)\text{.}$

4

When diving off a cliff from $x$ metres above the water, your speed as you hit the water is given by

\begin{equation*} s(x)=\sqrt{19.6x}\;\frac{\mathrm{m}}{\mathrm{sec}} \end{equation*}

Your last dive was from a height of 4 metres.

Use a linear approximation of $\Delta y$ to estimate how much faster you will be falling when you hit the water if you jump from a height of 5 metres.
A diver makes three jumps: the first is from $x$ metres, the second from $x+\Delta x$ metres, and the third from $x+2\Delta x$ metres, for some fixed positive values of $x$ and $\Delta x\text{.}$ Which is bigger: the increase in terminal speed from the first to the second jump, or the increase in terminal speed from the second to the third jump?

Subsubsection Exercises for § 3.4.7

Exercises — Stage 1

1

Let $f(x)=7x^2-3x+4\text{.}$ Suppose we measure $x$ to be $x_0 = 2$ but that the real value of $x$ is $x_0+\Delta x\text{.}$ Suppose further that the error in our measurement is $\Delta x = 1\text{.}$ Let $\Delta y$ be the change in $f(x)$ corresponding to a change of $\Delta x $ in $x_0\text{.}$ That is, $\Delta y = f\left(x_0+\Delta x\right)-f(x_0)\text{.}$

True or false: $\Delta y = f'(2)(1)=25$

2

Suppose the exact amount you are supposed to tip is $5.83, but you approximate and tip $6. What is the absolute error in your tip? What is the percent error in your tip?

3

Suppose $f(x)=3x^2-5\text{.}$ If you measure $x$ to be $10\text{,}$ but its actual value is $11\text{,}$ estimate the resulting error in $f(x)$ using the linear approximation, and then the quadratic approximation.

Exercises — Stage 2

4

A circular pen is being built on a farm. The pen must contain $A_0$ square metres, with an error of no more than 2%. Estimate the largest percentage error allowable on the radius.

5

A circle with radius 3 has a sector cut out of it. It's a smallish sector, no more than a quarter of the circle. You want to find out the area of the sector.

Suppose the angle of the sector is $\theta\text{.}$ What is the area of the sector?
Unfortunately, you don't have a protractor, only a ruler. So, you measure the chord made by the sector (marked $d$ in the diagram above). What is $\theta$ in terms of $d\text{?}$
Suppose you measured $d=0.7\text{,}$ but actually $d=0.68\text{.}$ Estimate the absolute error in your calculation of the area removed.

6

A conical tank, standing on its pointy end, has height 2 metres and radius 0.5 metres. Estimate change in volume of the water in the tank associated to a change in the height of the water from 50 cm to 45 cm.

Exercises — Stage 3

7

A sample begins with precisely 1 $\mu$g of a radioactive isotope, and after 3 years is measured to have 0.9 $\mu$g remaining. If this measurement is correct to within 0.05 $\mu$g, estimate the corresponding accuracy of the half-life calculated using it.

Subsubsection Exercises for § 3.4.8

Exercises — Stage 1

1

Suppose $f(x)$ is a function that we approximated by $F(x)\text{.}$ Further, suppose $f(10)=-3\text{,}$ while our approximation was $F(10)=5\text{.}$ Let $R(x)=f(x)-F(x)\text{.}$

True or false: $|R(10)| \leq 7$
True or false: $|R(10)| \leq 8$
True or false: $|R(10)| \leq 9$
True or false: $|R(10)| \leq 100$

2

Let $f(x)=e^x\text{,}$ and let $T_3(x)$ be the third-degree Maclaurin polynomial for $f(x)\text{,}$

\begin{equation*} T_3(x)=1+x+\frac{1}{2}x^2+\frac{1}{3!}x^3 \end{equation*}

Use Equation 3.4.33 to give a reasonable bound on the error $|f(2)-T_3(2)|\text{.}$ Then, find the error $|f(2)-T_3(2)|$ using a calculator.

3

Let $f(x)= 5x^3-24x^2+ex-\pi^4\text{,}$ and let $T_5(x)$ be the fifth-degree Taylor polynomial for $f(x)$ about $x=1\text{.}$ Give the best bound you can on the error $|f(37)-T(37)|\text{.}$

4

You and your friend both want to approximate $\sin(33)\text{.}$ Your friend uses the first-degree Maclaurin polynomial for $f(x)=\sin x\text{,}$ while you use the zeroth-degree (constant) Maclaurin polynomial for $f(x)=\sin x\text{.}$ Who has a better approximation, you or your friend?

Exercises — Stage 2

5

Suppose a function $f(x)$ has sixth derivative

\begin{equation*} f^{(6)}(x)=\dfrac{6!(2x-5)}{x+3}. \end{equation*}

Let $T_5(x)$ be the 5th-degree Taylor polynomial for $f(x)$ about $x=11\text{.}$

Give a bound for the error $|f(11.5)-T_5(11.5)|\text{.}$

6

Let $f(x)= \tan x\text{,}$ and let $T_2(x)$ be the second-degree Taylor polynomial for $f(x)$ about $x=0\text{.}$ Give a reasonable bound on the error $|f(0.1)-T(0.1)|$ using Equation 3.4.33.

7

Let $f(x)=\log (1-x)\text{,}$ and let $T_5(x)$ be the fifth-degree Maclaurin polynomial for $f(x)\text{.}$ Use Equation 3.4.33 to give a bound on the error $|f\left(-\frac{1}{4}\right)-T_5\left(-\frac{1}{4}\right)|\text{.}$

(Remember $\log x=\log_ex\text{,}$ the natural logarithm of $x\text{.}$)

8

Let $f(x)=\sqrt[5]{x}\text{,}$ and let $T_3(x)$ be the third-degree Taylor polynomial for $f(x)$ about $x=32\text{.}$ Give a bound on the error $|f(30)-T_3(30)|\text{.}$

9

Let

\begin{equation*} f(x)= \sin\left(\dfrac{1}{x}\right), \end{equation*}

and let $T_1(x)$ be the first-degree Taylor polynomial for $f(x)$ about $x=\dfrac{1}{\pi}\text{.}$ Give a bound on the error $|f(0.01)-T_1(0.01)|\text{,}$ using Equation 3.4.33. You may leave your answer in terms of $\pi\text{.}$

Then, give a reasonable bound on the error $|f(0.01)-T_1(0.01)|\text{.}$

10

Let $f(x)=\arcsin x\text{,}$ and let $T_2(x)$ be the second-degree Maclaurin polynomial for $f(x)\text{.}$ Give a reasonable bound on the error $\left|f\left(\frac{1}{2}\right)-T_2\left(\frac{1}{2}\right)\right|$ using Equation 3.4.33. What is the exact value of the error $\left|f\left(\frac{1}{2}\right)-T_2\left(\frac{1}{2}\right)\right|\text{?}$

Exercises — Stage 3

11

Let $f(x)=\log(x)\text{,}$ and let $T_n(x)$ be the $n$th-degree Taylor polynomial for $f(x)$ about $x=1\text{.}$ You use $T_n(1.1)$ to estimate $\log (1.1)\text{.}$ If your estimation needs to have an error of no more than $10^{-4}\text{,}$ what is an acceptable value of $n$ to use?

12

Give an estimation of $\sqrt[7]{2200}$ using a Taylor polynomial. Your estimation should have an error of less than 0.001.

13

Use Equation 3.4.33 to show that

\begin{equation*} \frac{4241}{5040}\leq\sin(1) \leq\frac{4243}{5040} \end{equation*}

14

In this question, we use the remainder of a Maclaurin polynomial to approximate $e\text{.}$

Write out the 4th degree Maclaurin polynomial $T_4(x)$ of the function $e^x\text{.}$
Compute $T_4(1)\text{.}$
Use your answer from 3.4.11.14.b to conclude $\dfrac{326}{120} \lt e \lt \dfrac{325}{119}\text{.}$

Subsubsection Further problems for § 3.4

Exercises — Stage 1

1 (✳)

Consider a function $f(x)$ whose third-degree Maclaurin polynomial is $4 + 3x^2 + \frac{1}{2}x^3\text{.}$ What is $f'(0)\text{?}$ What is $f''(0)\text{?}$

2 (✳)

Consider a function $h(x)$ whose third-degree Maclaurin polynomial is $1+4x-\dfrac{1}{3}x^2 + \dfrac{2}{3}x^3\text{.}$ What is $h^{(3)}(0)\text{?}$

3 (✳)

The third-degree Taylor polynomial of $h(x)$ about $x=2$ is $3 + \dfrac{1}{2}(x-2) + 2(x-2)^3\text{.}$

What is $h'(2)\text{?}$ What is $h''(2)\text{?}$

Exercises — Stage 2

4 (✳)

The function $f(x)$ has the property that $f(3)=2,\ f'(3)=4$ and $f''(3)=-10\text{.}$

Use the linear approximation to $f(x)$ centred at $x=3$ to approximate $f(2.98)\text{.}$
Use the quadratic approximation to $f(x)$ centred at $x=3$ to approximate $f(2.98)\text{.}$

5 (✳)

Use the tangent line to the graph of $y = x^{1/3}$ at $x = 8$ to find an approximate value for $10^{1/3}\text{.}$ Is the approximation too large or too small?

6 (✳)

Estimate $\sqrt{2}$ using a linear approximation.

7 (✳)

Estimate $\sqrt[3]{26}$ using a linear approximation.

8 (✳)

Estimate $(10.1)^5$ using a linear approximation.

9 (✳)

Estimate $\sin\left(\dfrac{101\pi}{100}\right)$ using a linear approximation. (Leave your answer in terms of $\pi\text{.}$)

10 (✳)

Use a linear approximation to estimate $\arctan(1.1)\text{,}$ using $\arctan 1 = \dfrac{\pi}{4}\text{.}$

11 (✳)

Use a linear approximation to estimate $(2.001)^3\text{.}$ Write your answer in the form $n/1000$ where $n$ is an integer.

12 (✳)

Using a suitable linear approximation, estimate $(8.06)^{2/3}\text{.}$ Give your answer as a fraction in which both the numerator and denominator are integers.

13 (✳)

Find the third--order Taylor polynomial for $f(x)=(1 - 3x)^{-1/3}$ around $x = 0\text{.}$

14 (✳)

Consider a function $f(x)$ which has $f^{(3)}(x)=\dfrac{x}{22-x^2}\text{.}$ Show that when we approximate $f(2)$ using its second degree Taylor polynomial at $a=1\text{,}$ the absolute value of the error is less than $\frac{1}{50}=0.02\text{.}$

15 (✳)

Consider a function $f(x)$ which has $f^{(4)}(x)=\dfrac{\cos(x^2)}{3-x}\text{.}$ Show that when we approximate $f(0.5)$ using its third-degree Maclaurin polynomial, the absolute value of the error is less than $\frac{1}{500}=0.002\text{.}$

16 (✳)

Consider a function $f(x)$ which has $f^{(3)}(x)=\dfrac{e^{-x}}{8+x^2}\text{.}$ Show that when we approximate $f(1)$ using its second degree Maclaurin polynomial, the absolute value of the error is less than $1/40\text{.}$

17 (✳)

By using an appropriate linear approximation for $f(x)=x^{1/3}\text{,}$ estimate $5^{2/3}\text{.}$
Improve your answer in 3.4.11.17.a by making a quadratic approximation.
Obtain an error estimate for your answer in 3.4.11.17.a (not just by comparing with your calculator's answer for $5^{2/3}$).

Exercises — Stage 3

18

The 4th degree Maclaurin polynomial for $f(x)$ is

\begin{equation*} T_4(x)=5x^2-9 \end{equation*}

What is the third degree Maclaurin polynomial for $f(x)\text{?}$

19 (✳)

The equation $y^4+xy=x^2-1$ defines $y$ implicitly as a function of $x$ near the point $x=2,\ y=1\text{.}$

Use the tangent line approximation at the given point to estimate the value of $y$ when $x=2.1\text{.}$
Use the quadratic approximation at the given point to estimate the value of $y$ when $x=2.1\text{.}$
Make a sketch showing how the curve relates to the tangent line at the given point.

20 (✳)

The equation $x^4+y+xy^4=1$ defines $y$ implicitly as a function of $x$ near the point $x=-1, y=1\text{.}$

Use the tangent line approximation at the given point to estimate the value of $y$ when $x=-0.9\text{.}$
Use the quadratic approximation at the given point to get another estimate of $y$ when $x=-0.9\text{.}$
Make a sketch showing how the curve relates to the tangent line at the given point.

21 (✳)

Given that $\log 10\approx 2.30259\text{,}$ estimate $\log 10.3$ using a suitable tangent line approximation. Give an upper and lower bound for the error in your approximation by using a suitable error estimate.

22 (✳)

Consider $f(x)=e^{e^x}\text{.}$

Give the linear approximation for $f$ near $x=0$ (call this $L(x)$).
Give the quadratic approximation for $f$ near $x=0$ (call this $Q(x)$).
Prove that $L(x) \lt Q(x) \lt f(x)$ for all $x \gt 0\text{.}$
Find an interval of length at most $0.01$ that is guaranteed to contain the number $e^{0.1}\text{.}$