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.
