The Art and Craft of Problem Solving

330 CHAPTER 9 CALCULUS

as desired. •

The tangent-line definition of the derivative stems from the formal definition of

the derivative as a limit. One of the first things you learned in your calculus class was

the definition

f'(a):= lim

f(x) -f(a)

= lim

f(a+h) -f(a)

x--->a.

x-a h--->O h

The fractions in the definition compute the slope of "secant lines" that approach the

tangent line in the limit. This suggests a useful, but less well-known, application of

the derivative, the tangent-line approximation to the function. For example, suppose

that f(3) = 2 and f'(3) = 4. Then

lim

f(3 +h) -f(3)

= 4.

h--->O h

Thus when h is small in absolute value, (/(3 +h) -f(3))/h will be close to 4; there­

fore,

f(3 +h) � f(3) +4h = 2+4h.

In other words, the function 1!(h) := (^2) +4h is the best linear approximation to f(3 +h),
in the sense that it is the only linear function 1!( h) that satisfies

I·

f(3+h)-1!(h)_

1m 0

h

-.
h--->O

In other words, 1!(h) is the only linear function that agrees with f(3 +h) and f'( (^3) +h)
when h =0.
In general, analyzing f(a + h) with its tangent-line approximation f(a) + hf'(a)
is very useful, especially when combined with other geometric information, such as
convexity.
Example 9.3.3 Prove Bernoulli's Inequality:

(1 +x)a 2: 1 + ax,

for x > -1 and a 2: 1, with equality when x = O.

Solution: For integer a, this can be proven by induction, and indeed, this was

Problem 2.3.33 on page 51. But induction won't work for arbitrary real a. Instead,

define f(u) := ua, and note that f'(u) = aua-I and J"(u) = a(a - 1 )ua-2• Thus f( 1) =

I,!, (1) = a and J" (u) >^0 as long as u >^0 (provided, of course, that a 2: 1). Thus the

graph y = f(u) is concave-up, for all u 2: 0 , as shown below.