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Homework 6.4.1: Constructing Antiderivatives

HW 6.4.1: #1, 2, 4, 7, 8, 9, 12, 14, 16, 17, 18, 19, 23

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6.4 SECOND FUNDAMENTAL THEOREM OF
CALCULUS
Suppose f is an elementary function, that is, a combination of constants, powers of x, sin x, cos x, ex, and ln
x. Then we have to be lucky to find an antiderivative F which is also an elementary function. But if we can’t
find F as an elementary function, how can we be sure that F exists at all? In this section we use the definite
integral to construct antiderivatives

.

Construction of Antiderivatives Using the Definite Integral
Consider the function . Its antiderivative, F, is not an elementary function, butwe would still
like to find a way of calculating its values. Assuming F exists, we know from the Fundamental Theorem of
Calculus that

()

Setting a = 0 and replacing b by x, we have

()
Suppose we want the antiderivative that satisfies F (0) = 0. Then we get

()
This is a formula for F. For any value of x, there is a unique value for F (x), so F is a function. For any fixed
x, we can calculate F (x) numerically. For example,

()
Notice that our expression for F is not an elementary function; we have created a new function using the
definite integral. The next theorem says that this method of constructing antiderivatives works in general.
This means that if we define F by

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()
then F must be an antiderivative of f.
Theorem 6.2: Construction Theorem for Antiderivatives
(Second Fundamental Theorem of Calculus) If f is a continuous function on an interval, and if a is any
number in that interval, then the function F defined on the interval as follows is an antiderivative of f:

()

Proof
Our task is to show that F, defined by this integral, is an antiderivative of f. We want to show that F′(x) =
f (x). By the definition of the derivative,

()
To gain some geometric insight, let’s suppose f is positive and h is positive. Then we can visualize

()
as areas, which leads to representing

()
as a difference of two areas. From Figure 6.30, we see that F (x + h) − F (x) is roughly the area of a
rectangle of height f (x) and width h (shaded darker in Figure 6.30), so we have

()
hence

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()
More precisely, we can use Theorem 5.4 on comparing integrals to conclude that

()
where m is the greatest lower bound for f on the interval from x to x + h and M is the least upper bound on
that interval. (See Figure 6.31.) Hence

()
so

()
Since f is continuous, both m and M approach f (x) as h approaches zero. Thus

()
Thus both inequalities must actually be equalities, so we have the result we want:

()
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Figure 6.30: F (x + h) − F (x) is area of roughly rectangular region

Figure 6.31: Upper and lower bounds for F (x + h) − F (x)

Relationship Between the Construction Theorem and the Fundamental
Theorem of Calculus
If F is constructed as in Theorem 6.2, then we have just shown that F′ = f. Suppose G is any other
antiderivative of f, so G′ = f, and therefore F′ = G′. Since the derivative of F − G is zero, the Constant
Function Theorem tells us that F − G is a constant, so F (x) = G(x) + C.
Since we know F (a) = 0 (by the definition of F), we can write

()

This result, that the definite integral can be evaluated using any antiderivative of f, is the (First)

Fundamental Theorem of Calculus. Thus, we have shown that the First Fundamental Theorem of Calculus
can be obtained from the Construction Theorem (the Second Fundamental Theorem of Calculus).

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Using the Construction Theorem for Antiderivatives
The construction theorem enables us to write down antiderivatives of functions that do not have elementary
antiderivatives. For example, an antiderivative of (sin x)/x is

()
Notice that F is a function; we can calculate its values to any degree of accuracy.1 You may notice that the
integrand, (sin t)/t, is undefined at t = 0; such improper integrals are treated in more detail in Chapter 7. This
function already has a name: it is called the sine-integral, and it is denoted Si(x).
Example 1
Construct a table of values of Si(x) for x = 0, 1, 2, 3.
Solution

Using numerical methods, we calculate the values of given in Table 6.2. Since the

integrand is undefined at t = 0, we took the lower limit as 0.00001 instead of 0.
Table6.2A table of
values of Si(x)

x 0 1 2 3
Si(x)00.951.611.85

The reason the sine-integral has a name is that some scientists and engineers use it all the time (for example,
in optics). For them, it is just another common function like sine or cosine. Its derivative is given by

()

Example 2
Find the derivative of x Si(x).
Solution
Using the product rule,

()
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Exercises and Problems for Section 6.4
EXERCISES
1.

For x = 0, 0.5, 1.0, 1.5, and 2.0, make a table of values for .

2.
Assume that F′(t) = sin t cos t and F (0) = 1. Find F (b) for b = 0, 0.5, 1, 1.5, 2, 2.5, and 3.

3.

(a)

Continue the table of values for for x = 4 and x = 5.

(b)
Why is Si(x) decreasing between x = 4 and x = 5?

In Exercises 4-6, write an expression for the function, f (x), with the given properties.
4.
f′(x) = sin(x2) and f (0) = 7

5.
f′(x) = (sin x)/x and f (1) = 5

6.
f′(x) = Si(x) and f (0) = 2

In Exercises 7-10, let . Graph F (x) as a function of x.

7.

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8.

9.

10.

Find the derivatives in Exercises 11-16.
11.

12.

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13.

14.

15.

16.

PROBLEMS
17.

Find intervals where the graph of is concave up and concave down.

18.
Use properties of the function f (x) = xe−x to determine the number of values x that make F (x) = 0, given

.

For Problems 19-21, let .

19.
Approximate F (x) for x = 0, 0.5, 1, 1.5, 2, 2.5.

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20.
Using a graph of F′(x), decide where F (x) is increasing and where F (x) is decreasing for 0 ≤ x ≤ 2.5.

21.
Does F (x) have a maximum value for 0 ≤ x ≤ 2.5? If so, at what value of x does it occur, and approximately
what is that maximum value?

22.

Use Figure 6.32 to sketch a graph of . Label the points x1, x2, x3.

Figure 6.32

23.
The graph of the derivative F′ of some function F is given in Figure 6.33. If F (20) = 150, estimate the
maximum value attained by F.

Figure 6.33

24.

Figure 6.34
(a)

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Let , with f in Figure 6.24. Find the x-value at which the maximum value of F occurs.

(b)
What is the maximum value of F?

25.

Let . Using Figure 6.35, find

Figure 6.35
(a)
g(0)

(b)
g′(1)

(c)
The interval where g is concave up.

(d)
The value of x where g takes its maximum on the interval 0 ≤ x ≤ 8.

26.

Let .

(a)
Evaluate .

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(b)
Draw a sketch to explain geometrically why the answer to part a is correct.

(c)
For what values of x is F (x) positive? negative?

27.

Let for x ≥ 2.

(a)
Find F′(x).

(b)
Is F increasing or decreasing? What can you say about the concavity of its graph?

(c)
Sketch a graph of F (x).

In Problems 28-29, find the value of the function with the given properties.
28.
F (1), where and F (0) = 2

29.
G(−1), where G′(x) = cos(x2) and G(0) = −3

In Problems 30-33, estimate the value of each expression, given and

. Table 6.3 gives values for q(x), a function with negative first and second

derivatives. Are your answers under- or overestimates?
Table6.3

x 0.00.10.20.30.40.5
q(x)5.35.24.94.53.93.1
30.
w(0.4)

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31.
v(0.4)

32.
w′(0.4)

33.
v′(0.4)

In Problems 34-37, use the chain rule to calculate the derivative.
34.

35.

36.

37.

In Problems 38-41, find the given quantities. The error function, erf(x), is defined by

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()
38.

39.

40.

41.

42.
In year x, the forested area in hectares of an island is

()

where r(t) has only one zero, at t = 40. Let r′(40) < 0, f (40) = 1170 and .

(a)
What is the maximum forested area?

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(b)
How large is the forested area in year t = 30?

43.

Let and , where F (3) = 5 and G(5) = 7. Find each of the

following.
(a)
G(0)

(b)
F (0)

(c)
F (5)

(d)
k where G(x) = F (x) + k

Strengthen Your Understanding
In Problems 44-46, explain what is wrong with the statement.

44.

45.

has a local minimum at x = 0.

46.

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For the function f (x) shown in Figure 6.36, has a local minimum at x = 2.

Figure 6.36

In Problems 47-48, give an example of:
47.
A function, F (x), constructed using the Second Fundamental Theorem of Calculus, such that F is a
nondecreasing function and F (0) = 0.

48.
A function G(x), constructed using the Second Fundamental Theorem of Calculus such that G is concave up
and G(7) = 0.

Are the statements in Problems 49-54 true or false? Give an explanation for your answer.
49.
Every continuous function has an antiderivative.

50.

is an antiderivative of sin(x2).

51.

If , then .

52.

If , then F (x) must be increasing.

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53.

If and , then F (x) = G(x) + C.

54.

If and , then .

Additional Problems
AP55.

Let where . At x = 3, is G(x) increasing or decreasing? Is its graph

concave up or concave down?

AP56.

Suppose that f (x) is a continuous function and for all a and b.

(a)

Show that for all x.

(b)
Show that f (x) = 0 for all x.

AP57.

Let

(a)
Evaluate R(0) and determine if R is an even or an odd function.

(b)
Is R increasing or decreasing?

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(c)
What can you say about concavity?

(d)
Sketch a graph of R(x).

(e)
Show that limx → ∞(R(x)/x2) exists and find its value.

AP58.

Let where Figure AP6.37 shows the graph of y = s(x). Consider the eleven x-values x

= a, x = 0, x = b, x = c, …, x = j. For each statement (I)-(IV), which of these eleven x-values satisfy the
statement?

I.S(x) is positive.
II.S(x) is a local minimum.
III.S(x) is a point of inflection.
IV.S(x) is the maximum value for S on [a, j].

Figure AP6.37

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