### Math 242 A Homework #1 Spring, 2021

Assigned February 10. Due February 19. Point values are as marked.

1. (20) Define

Show that 2^{(n) }(x) exists for all x and all n, and that f(n)
(0) = 0 for all n. You may assume standard calculus facts about exponentials, polynomials, and rational functions.

2. (15) Let f : [a, b] → **R** be continuous, with a < b, and suppose that f′
exists (finite or infinite) everywhere on (a, b). Suppose that

and

both exist as real numbers. Show that f′+(a) and f′_(b) both exist as real numbers, with A = f′+(a) and B = f′_(b).

3. (15) Let f : **R** → **R** have a derivative (finite or infinite) at all points, and suppose that

and

both exist as real numbers. Show that the second limit equals 0.

4. (15) Let f : **R** → **R** have a derivative (finite or infinite) at all points, and suppose that

Show that the limit

exists and equals 0.

5. (20) Suppose f : [0, 1] → **R** is continuous, f(0) = 0, and f′(x) exists and is finite for all x ∈ (0, 1). Suppose furthermore that f′(x) is an *increasing *function on (0, 1). Show that f(x)/x is also increasing on (0, 1). Give an example to show that this conclusion can fail if we remove the hypothesis that f(0) = 0.

6. (15) Let f : (0, 1) → **R** have a finite derivative everywhere in (0, 1), with |f′(x)| ≤ 1 ?x ∈ (0, 1). Show that

and

both exist as real numbers.

2021-02-20