The Art and Craft of Problem Solving

340 CHAPTER 9 CALCULUS

four distinct points.

9.3. 16 Convert the statement "the tangent line is

the best linear approximation to the function" into

a rigorous statement that uses Iittle-oh notation (see

page 319).

9.3. 17 Let f(x) be a differentiable function that satis­

fies

f(x + y) = f(x)f(y)

for all x,y E JR. If f' (0) = 3, find f(x).

9.3. 18 Let f(x) be a differentiable function that satis­

fies

f(xy) = f(x) + f(y)

for all x,y > O. If f' (1) = 3, find f(x).

9.3. 19 (Putnam 194 6) Let f(x) := ax 2 +bx+c, where

a,b,c E JR. If If(x) I :S 1 for Ixl :S I, prove that If' (x) I :S

4 for Ixl :S 1.

9.3.20 More About the Mean Va lue Theorem.

(a) The "proof' of the mean value theorem on

page 332 was simply to "tilt" the picture for

Rolle's theorem. Prove the mean value theo­

rem slightly more rigorously now, by assuming

the truth of Rolle's theorem, and defining a new

function in such a way that Rolle's theorem ap­

plied to this new function yields the mean value

theorem. Use the tilting-picture idea as your

guide.

(b) If you succeeded in (a), you may still grumble

that you did merely an algebra exercise, really

nothing new, and certainly achieved no insight

better than tilting the picture. This is true, but

the algebraic method is easy to generalize. Use

it to prove the generalized mean value theo­

rem, which involves two functions:

Let f(x) and g(x) be continuous

on [a,b] and diff erentiable on (a,b).

Then there is a point u E (a,b) such

that

!' (u)(g(b) -g(a)) = g' (u)(f(b) -f(a)).

(c) The regular mean value theorem is a special

case of the generalized mean value theorem.

Explain why.

(d) It is still worthwhile to understand the gener­

alized mean value theorem pictorially. Draw a

picture to illustrate what it says. Can you de­

velop a pictorial proof, similar to the one we

did for the mean value theorem?

9.3.21 In Example 9.2.3, you were asked to show that

�n + 1 -� = O(n- 2 / 3 ). Presumably you have done

so, most likely using algebra. Show it again, but this

time, use the Mean Value Theorem!

9.3.22 Use the generalized mean value theorem

(9.3.20) to prove the following variant of L'Hopital's

Rule:

Let f(x) and g(x) be diff erentiable on an

open interval containing x = a. Suppose

also that

Then

x-+a lim f(x) = x--+a lim g(x) = O.

lim

f(x)

= lim!

,(x)

x-+a g(x) x-+a g' (x) ,

provided that neither g(x) nor g' (x)

equal zero for xi-a.

9.3.23 The Mean Va lue Theorem for Integrals. This

theorem states the following:

Let f(x) be continuous on (a, b). Then

there is a point u E (a, b) at which

f(u)(b - a) = lb f(x)dx.

(a) Draw a picture to see why this theorem is plau­

sible, in fact "obvious."

(b) Use the regular mean value theorem to prove it

(define a function cleverly, etc.).

9.3.24 (Putnam 1987) Evaluate

f4 v'ln(9-x)dx

(^12) yln(9-x)+yln(x+3)
9.3.25 (Putnam 1946) Let f : JR --+ JR have a continu­
ous derivative, with f(O) = 0, and If'(x)1 :S If(x) I for
all x. Show that f is constant.
9.3.26 Find and prove (using induction) a nice for­
mula for the nth derivative of the product f(x)g(x).
9.3.27 (Putnam 1993) The horizontal line y = c inter­
sects the curve y = 2x -3� in the first quadrant as in
the figure. Find c so that the areas of the two shaded
regions are equal.