James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

392 Chapter 5 Integrals
of x at which tsxd starts to decrease. [Unlike the integral in Problem 2, it is impossible
to evaluate the integral defining t to obtain an explicit expression for tsxd.]
(c) Use the integration command on your calculator or computer to estimate ts0.2d,
ts0.4d, ts0.6d, . . . , ts1.8d, ts2d. Then use these values to sketch a graph of t.
(d) Use your graph of t from part (c) to sketch the graph of t9 using the interpretation of
t9sxd as the slope of a tangent line. How does the graph of t9 compare with the graph
of f ?
4. Suppose f is a continuous function on the interval fa, bg and we define a new function t
by the equation
tsxd − y x
f std dt
a
Based on your results in Problems 1–3, conjecture an expression for t9sxd.
y
y=f(t)
The Fundamental Theorem of Calculus is appropriately named because it establishes a
con nection between the two branches of calculus: differential calculus and integral
calculus. Differential calculus arose from the tangent problem, whereas integral calculus
arose from a seemingly unrelated problem, the area problem. Newton’s mentor at
Cambridge, Isaac Barrow (1630 –1677), discovered that these two problems are actually
closely related. In fact, he realized that differentiation and integration are inverse
processes. The Fundamental Theorem of Calculus gives the precise inverse relationship
between the derivative and the integral. It was Newton and Leibniz who exploited this
relationship and used it to develop calculus into a systematic mathema tical method. In
particular, they saw that the Fundamental Theorem enabled them to compute areas and
integrals very easily without having to compute them as limits of sums as we did in Sections
5.1 and 5.2.
The first part of the Fundamental Theorem deals with functions defined by an equation
of the form
area=©
1
tsxd − y x
f std dt
a
0 a
x
FIGURE 1
y
2
y=f(t)
1
0 1 2 4
b
t
t
where f is a continuous function on fa, bg and x varies between a and b. Observe that t
depends only on x, which appears as the variable upper limit in the integral. If x is a fixed
number, then the integral y x a
f std dt is a definite number. If we then let x vary, the number
y x a
f std dt also varies and defines a function of x denoted by tsxd.
If f happens to be a positive function, then tsxd can be interpreted as the area under the
graph of f from a to x, where x can vary from a to b. (Think of t as the “area so far”
function; see Figure 1.)
Example 1 If f is the function whose graph is shown in Figure 2 and
tsxd − y x 0
f std dt, find the values of ts0d, ts1d, ts2d, ts3d, ts4d, and ts5d. Then sketch a
rough graph of t.
FIGURE 2
SOLUTION First we notice that ts0d − y 0 0
f std dt − 0. From Figure 3 we see that ts1d is
the area of a triangle:
ts1d − y 1
f std dt − 1 2 s1 ? 2d − 1
0
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.