## Description

Exercises 5.2.3 (c), 5.3.1, 5.3.2, 6.2.1, 6.2.2 (a), 6.3.1, 6.4.5, 6.5.3, 6.6.5, 6.6.6

## Exercise 5.2.3.

(a) Use Definition 5.2.1 to produce the proper formula for the derivative of h(x) = 1/x.

(b) Combine the result in part (a) with the Chain Rule (Theorem 5.2.5) to supply a proof for part (iv) of Theorem 5.2.4.

(c) Supply a direct proof of Theorem 5.2.4 (iv) by algebraically manipulat- ing the difference quotient for (f/g) in a style similar to the proof of Theorem 5.2.4 (iii).

## Exercise 5.3.1.

Recall from Exercise 4.4.9 that a function f A→ R is Lipschitz on A if there exists an M> 0 such that

for all xy in A.

|f(x) f(y) |

<M

x-y

(a) Show that if ƒ is differentiable on a closed interval [a, b] and if f’ is con- tinuous on [a, b], then ƒ is Lipschitz on [a, b].

(b) Review the definition of a contractive function in Exercise 4.3.11. If we add the assumption that f'(x) < 1 on [a, b], does it follow that ƒ is contractive on this set?

## Exercise 5.3.2.

Let f be differentiable on an interval A. If f'(x) 0 on A, show that ƒ is one-to-one on A. Provide an example to show that the converse statement need not be true.

## Exercise 6.2.2.

(a) Define a sequence of functions on R by

fn(2)

={

if x = 1, 4, 4. 0 otherwise

1

23

and let ƒ be the pointwise limit of fn.

Is each fn continuous at zero? Does fnf uniformly on R? Is f continuous at zero?

## Exercise 6.3.1.

Consider the sequence of functions defined by

In(x):

xn

n

(a) Show (gn) converges uniformly on [0, 1] and find g = lim gn. Show that g is differentiable and compute g'(x) for all x € [0, 1].

(b) Now, show that (g) converges on [0, 1]. Is the convergence uniform? Set h = lim g, and compare h and g’. Are they the same?

## Exercise 6.4.5.

(a) Prove that

xn

h(x) =

x4

n2

+

4

9

16

is continuous on [-1, 1].

(b) The series

f(x)=

x2

23

= x +

+ +

n

2 3

n=1

converges for every x in the half-open interval [−1, 1) but does not converge when = 1. For a fixed to € (-1, 1), explain how we can still use the Weierstrass M-Test to prove that ƒ is continuous at xo.

## Exercise 6.6.5.

(a) Generate the Taylor coefficients for the exponential func- tion f(x) = e, and then prove that the corresponding Taylor series con- verges uniformly to ea on any interval of the form [-R, R].

(b) Verify the formula f'(x) = e.

(c) Use a substitution to generate the series for e-, and then informally calculate ee by multiplying together the two series and collecting common powers of x.

## Exercise 6.6.6.

Review the proof that g'(0) = 0 for the function

9(x) = { {

e-1/x2 0

for x 0, for x = 0.

introduced at the end of this section.

(a) Compute g'(x) for x 0. Then use the definition of the derivative to find g”(0).

(b) Compute g'(x) and g”(x) for x 0. Use these observations and in- vent whatever notation is needed to give a general description for the nth derivative g(n) (x) at points different from zero.

(c) Construct a general argument for why g(n) (0) = 0 for all nЄ N.