Section 4.1 Introduction to Antiderivatives
Subsection 4.1.1 Introduction to Antiderivatives
So far in the course we have learned how to determine the rate of change (i.e. the derivative) of a given function. That is
Along the way we developed an understanding of limits, which allowed us to define instantaneous rates of change — the derivative. We then went on to develop a number of applications of derivatives to modelling and approximation. In this last section we want to just introduce the idea of antiderivatives. That is
For example — say we know that
and we want to find From our previous experience differentiating we know that derivatives of polynomials are again polynomials. So we guess that our unknown function is a polynomial. Further we know that when we differentiate we get — multiply by the exponent and reduce the exponent by 1. So to end up with a derivative of we need to have differentiated an But so we need to divide both sides by 3 to get the answer we want. That is
However we know that the derivative of a constant is zero, so we also have
and
At this point it will really help the discussion to give a name to what we are doing.
Notice the use of the indefinite article there — an antiderivative. This is precisely because we can always add or subtract a constant to an antiderivative and when we differentiate we’ll get the same answer. We can write this as a lemma, but it is actually just Corollary 2.13.13 (from back in the section on the mean-value theorem) in disguise.
Lemma 4.1.2.
Because of this lemma we typically write antiderivatives with “ ” tacked on the end. That is, if we know that then we would state that the antiderivative of is
where this “ ” is there to remind us that we can always add or subtract some constant and it will still be an antiderivative of Hence the antiderivative of is
Similarly, the antiderivative of is
and for it is
This last one is tricky (at first glance) — but we can always check our answer by differentiating.
Now in order to determine the value of we need more information. For example, we might be asked
We are given the derivative and one piece of additional information and from these two facts we need to find the original function. From our work above we know that
and we can find from the other piece of information
Hence and so
Finding antiderivatives of polynomials is generally not too hard. We just need to use the rule
Of course this breaks down when In order to find an antiderivative for we need to remember that and more generally that See Example 2.10.4. So
Note however, that both and are not defined at If you are interested, for example, in functions that are defined all then, for any constants and
obeys for all However, typically, in applications, either all of the ’s of interest are strictly positive, or all of the ’s of interest are negative. In both of those cases, the general is So, typically, in tables of antiderivatives, the antiderivative of is listed as
Example 4.1.3. Antiderivative of .
Let Then the antiderivative of is
Now to check we should differentiate and hopefully we get back to where we started
In your next calculus course you will develop a lot of machinery to help you find antiderivatives. At this stage about all that we can do is continue the sort of thing we have done. Think about the derivatives we know and work backwards. So, for example, we can take a list of derivatives
and flip it upside down to give the tables of antiderivatives.
Here is just a constant — any constant. But we can do a little more; clean up by dividing by and then replacing by Similarly we can tweak by multiplying by
Here are a couple more examples.
Example 4.1.4. Antiderivatives of and .
Consider the functions
Find their antiderivatives.
Solution The first one we can almost just look up our table. Let be the antiderivative of then
When we differentiate to check things, we get a factor of two coming from the chain rule. Hence to compensate for that we multiply by
Differentiating this shows that we have the right answer.
Similarly, if we use to denote the antiderivative of then it appears that is nearly To get this extra factor of we need to substitute So we try
Differentiating this gives us
Hence we should multiply by This gives us
We can then check that this is, in fact, correct just by differentiating.
Now let’s do a more substantial example.
Example 4.1.5. Position as antiderivative of velocity.
Suppose that we are driving to class. We start at at time Our velocity is The class is at When do we get there?
Solution Let’s denote by our position at time We are told that
We have to determine and then find the time that obeys Now armed with our table above we know that the antiderivative of is just We can check this:
We can then get the factor of by multiplying both sides of the above equation by 50:
And of course, this is just an antiderivative of to write down the general antiderivative we just add a constant
Since this antiderivative is
To determine we make use of the other piece of information we are given, namely
Substituting this in gives us
Hence we must have and so
Now that we have our position as a function of time, we can determine how long it takes us to arrive there. That is, we can find the time so that
Recalling our special triangles, we see that
The example below shows how antiderivatives arise naturally when studying differential equations.
Example 4.1.6. Theorem 3.3.2 revisited..
Back in Section 3.3 we encountered a simple differential equation, namely equation 3.3.1. We were able to solve this equation by guessing the answer and then checking it carefully. We can derive the solution more systematically by using antiderivatives.
Recall equation 3.3.1:
where is the amount of radioactive material at time and we assume Take this equation and divide both sides by to get
At this point we should think that the left-hand side is familiar. Now is a good moment to look back at logarithmic differentiation in Section 2.10.
1
Well — perhaps it is better to say “notice that”. Let’s not make this a moral point.
The left-hand side is just the derivative of
So to solve this equation, we are really being asked to find all functions having derivative That is, we need to find all antiderivatives of Of course that is just Hence we must have
and then taking the exponential of both sides gives
where This is precisely Theorem 3.3.2.
The above is a small example of the interplay between antiderivatives and differential equations.
Here is another example of how we might use antidifferentiation to compute areas or volumes.
Example 4.1.7. Volume of a cone.
We know (especially if one has revised the material in the appendix and Appendix B.5.2 in particular) that the volume of a right-circular cone is
where is the height of the cone and is the radius of its base. Now, the derivation of this formula given in Appendix B.5.2 is not too simple. We present an alternate proof here that uses antiderivatives.
Consider cutting off a portion of the cone so that its new height is (rather than ). Call the volume of the resulting smaller cone We are going to determine for all including by first evaluating and (which is obviously ).
Call the radius of the base of the new smaller cone (rather than ). By similar triangles we know that
Now keep and fixed and consider cutting off a little more of the cone so its height is When we do so, the radius of the base changes from to and again by similar triangles we know that
The change in volume is then
Of course if we knew the formula for the volume of a cone, then we could compute the above exactly. However, even without knowing the volume of a cone, it is easy to derive upper and lower bounds on this quantity. The piece removed has bottom radius and top radius Hence its volume is bounded above and below by the cylinders of height and with radius and respectively. Hence
since the volume of a cylinder is just the area of its base times its height. Now massage this expression a little
The middle term now looks like a derivative; all we need to do is take the limit as
The rightmost term is independent of and so is just In the leftmost term, as we must have that Hence the leftmost term is just Then by the squeeze theorem (Theorem 1.4.18) we know that
But we know that
so
Now we can antidifferentiate to get back to
To determine notice that when the volume of the cone is just zero and so Thus
and so when we are left with
as required.
Exercises 4.1.2 Exercises
Exercises — Stage 1 .
Exercises — Stage 2 .
3.
Find the antiderivative of
4.
Find the antiderivative of
5.
Find the antiderivative of
6.
Find the antiderivative of
7.
Find the antiderivative of
8.
Find the antiderivative of
9.
Find the antiderivative of
10.
Find the antiderivative of
11.
Find the antiderivative of
12.
Find the antiderivative of
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14.
15.
16.
17.
Suppose a population of bacteria at time (measured in hours) is growing at a rate of individuals per hour. Starting at time how long will it take the initial colony to increase by 300 individuals?
18.
19.
Exercises — Stage 3 .
For Questions 4.1.2.21 through 4.1.2.26, you are again asked to find the antiderivatives of certain functions. In general, finding antiderivatives can be extremely difficult--indeed, it will form the main topic of next semester’s calculus course. However, you can work out the antiderivatives of the functions below using what you’ve learned so far about derivatives.
20. (✳).
Let and Find the derivative of and simplify your answer. What does the answer imply about the relation between and
21.
Find the antiderivative of
22.
Find the antiderivative of
23.
Find the antiderivative of
24.
Find the antiderivative of
25.
Find the antiderivative of
26.
Find the antiderivative of