AMA3007-Assignment 3 Solved

Exercise 5.2.3. (a) Use Definition 5.2.1 to produce the proper formula for the derivative of h@) l/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 manipulating 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          y in A.

(a)    Show that if f is differentiable on a closed interval [a, b] and if f' is continuous on [a, b], then f is Lipschitz on [a, b].

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

Exercise 5.3.2. Let f be differentiable on an interval A. If f # 0 on A, show that f 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

 1 if x = 1 

0 otherwise and let f be the pointwise limit of fn.

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

Exercise 6.3.1. Consider the sequence of functions defined by

 

n

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

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

Exercise 6.4.5.    (a) Prove that

 

is continuous on [—1, 1].

(b) The series

 

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

Exercise 6.6.5. (a) Generate the Taylor coeffcients for the exponential function = ex , and then prove that the corresponding Taylor series converges uniformly to ex on any interval of the form [—R, R].

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

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

Exercise 6.6.6. Review the proof that g' (0) = O for the function

 —1/x2 for X 0, for = 0.

introduced at the end of this section.

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

(b)    Compute g" (x) and for 0. Use these observations and invent whatever notation is needed to give a general description for the nth derivative at points different from zero.

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