FDWK_3ed_Ch05_pp262-319

Section 5.4 Fundamental Theorem of Calculus 297
EXAMPLE 4 Constructing a Function with a Given
Derivative and Value
Find a function y f x with derivative
dy
tan x d x
that satisfies the condition f 3 5.
SOLUTION
The Fundamental Theorem makes it easy to construct a function with derivative tan x:
y x
3
tan tdt.
Since y3 0, we have only to add 5 to this function to construct one with derivative
tan x whose value at x 3 is 5:
f x x
tan tdt 5. Now try Exercise 25.
3
Although the solution to the problem in Example 4 satisfies the two required conditions,
you might question whether it is in a useful form. Not many years ago, this form might
have posed a computation problem. Indeed, for such problems much effort has been
expended over the centuries trying to find solutions that do not involve integrals. We will
see some in Chapter 6, where we will learn (for example) how to write the solution in
Example 4 as
y ln c os
3
cos
x 5.
However, now that computers and calculators are capable of evaluating integrals, the form
given in Example 4 is not only useful, but in some ways preferable. It is certainly easier to
find and is always available.
Graphing the Function a
xft dt
Consider for a moment the two forms of the function we have just been discussing,
Fx x
tan tdt 5 and F x ln c os
3
cos
x 5.
3
With which expression is it easier to evaluate, say, F4 From the time of Newton almost
to the present, there has been no contest: the expression on the right. At least it provides
something to compute, and there have always been tables or slide rules or calculators to
facilitate that computation. The expression on the left involved at best a tedious summing
process and almost certainly an increased opportunity for error.
Today we can find F4 from either expression on the same machine. The choice is
between NINT tan x, x,3,4 5 and ln abscos3cos4 5. Both calculations give
5.415135083 in approximately the same amount of time.
We can even use NINT to graph the function. This modest technology feat would have
absolutely dazzled the mathematicians of the 18th and 19th centuries, who knew how the
solutions of differential equations, such as dydx tan x, could be written as integrals, but