Assignment 4-

1. Using the definition of the derivative, determine the derivative of the function f(x) = √(2x + 1), at x = 1.

2. Suppose we know that f(x) is differentiable at x = a. For each of the following, determine if the statement is true or false. If true, explain why. If false, give a counterexample.

(a) f'(a) = lim_h→a(f(h)-f(a)/h-a).

(b) f'(a) = lim_h→0(f(a)-f(a-h)/h).

(c) f'(a) = lim_h→0(f(a+2h)-f(a)/h).

3. Let f(x) = |x|³. Is f(x) differentiable everywhere? If so, is f'(x) differentiable everywhere? If so, is f''(x) differentiable everywhere?

4. Show that each of the following statements are true.

(a) There do not exist functions f(x) and g(x), both differentiable everywhere, such that x = f(x)g(x), and f(0) = g(0) = 0.

(b) If f(x) is differentiable everywhere, and f(x) is an odd function, then f'(x) is an even function. [Recall: We say g(x) is an odd function if g(-x) = -g(x) for all x, and we say g(x) is an even function if g(-x) = g(x) for every x.]

5. (a) Let f(x) = √(1- √2(√3-x)). Find the domain of f(x), and compute f'(x).

(b) Let f(x) = xⁿ/1 - x.

Find, and prove, a general formula for f⁽ⁿ⁾(x) in terms of n, when n is a positive integer.