You need to explain all reasons in detail and show all of your work on
the answer sheet; otherwise you will get NO credits. If you used any theorem
in textbook or proved in class, state it carefully and explicitly.
I. A. (10%) Show that the function f(x) = ｜x｜ is comtinuous at 0 but
not differentiable at 0.
B. (10%) If f is differentiable at x, show that f is continuous at x.
II. A. (10%) Let f and g be differentiable functions such that f'(x) = g(x),
and let
T(x) = [f(x)]^2 + [g(x)]^2
find T'(x).
B. (10%) Let f be a differentiable function. Use the chain rule to show
that:
(a) if f is even, then f' is odd.
(b) if f is odd, then f' is even.
III. A. (10%) Set f(x) = x^(-1), a = -1, b = 1. Verify that there is no
number c for which
f(b) - f(a)
f'(c) = ────────
b - a
Explain how this does not violate the mean-value theorem.
B. (10%) Let f be differentiable on (a,b) and continuous on [a,b].
Prove that if there is a constant M such that f'(x) <= M
for all x∈(a,b), then
f(b) <= f(a) + M(b-a).
IV. A. (10%) Find f given that f'(x) = 6x^2-7x-5 for all real x and f(2) = 1.
B. (10%) Given
｜ x+2, x < 0
｜(x-1)^2, 0 < x < 3
｜ 8-x, 3 < x < 7
f(x) = ｜ 2x-9, 7 < x
｜ 6, x = 0, 3, 7
Find the intervals on which f increases and the intervals on
which f decreases.
V. (20%) Sketch the graph of f(x) = (1/4)x^4 - 2x^2 + 7/4 on [-5,5].
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