UMC Advanced Calculus Sequences Limits & Continuous Functions Exam Practice

In Questions 1-5 prove your answer. In this test you can usewithout proofs theorems that were proved in the lectures or in thebook, just give a reference.

1.(10 pts) Let f be a function on [?1, 1]. Two of the following state-ments, if combined, imply that f?(0) = 0. Which two statements?

A: limx?0 f(x) = 0;B:Thesequence{n(f(n1)?f(0))}?n=1 haslimitzero;C: f is differentiable at zero;
D: f is uniformly continuous on [?1, 1].

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2. (10 pts) Suppose that f, g are continuous functions on [0, 1], dif-ferentiable on (0, 1), g?(x) ?= 0 for all x ? (0, 1), and

f(0)=1, f(1)=2, g(0)=3, g(1)=1.
It is known that three of the following statements are true, while one

is not. Which one is not true? Explain your answer.

A: f ? g is an increasing function on (0, 1);B: f + g is a decreasing function on (0, 1);C: f?(x)g?(x) > 0 for every x ? (0,1);
D: f?(x)g?(x) < 0 for every x ? (0,1);

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3. (10 pts) Which of the following polynomials are increasing in aneighborhood of zero? Prove your answer.

A: 100×7 + 2×3 + x2B: 5×14 ?7×7 ?x5C: x10 +2×6 +4x3D: 4×9 ?3×8 ?5×6

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4.(10 pts) Which of the following inequalities are correct? Prove youranswer. You are not allowed to use a calculator for this problem.

A: 23/2 + 63/2 + 73/2 ? 3 · 53/2;
B: 2?3/2 + 6?3/2 + 7?3/2 < 3 · 5?3/2;

C:e2x +e6x +e7x ?3e5x foreveryx?R;D:e?2x +e?6x +e?7x <3e?5x foreveryx?R.Hint: If f is a convex function then

??x1 +…+xn?? f(x1)+…+f(xn)fn?n.

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5. (10 pts) Let f be a 1-1 continuous function on [0, 2]. Which threeof the following statements combined imply that f is differentiable at1 and f?(1) = 81? Prove your answer.

A:f(0)=3, f(2)=6;
B: f(1) = 8;
C: f?1 is differentiable at 5 and (f?1)?(5) = 8;D: f(1) = 5.

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6. (10 pts) (i) Suppose that f is a differentiable function everywhereon R, and f?(x) = 3 for every x ? R. Prove that f(x) = 3x+b, whereb is a constant.

(ii) Suppose that f is a differentiable positive function on R suchthat

f?(x) = 3f(x)for every x ? R. Prove that

f(x) = Ce3x,where C > 0 is an arbitrary positive constant.

7. (10 pts). Prove that for every polynomial
P(x)=a0 +a1x+a2x2 +….+anxn, an ?=0

there exists M ? R so that P is strictly increasing or strictly decreasingon [M, ?).

Hint: Use a theorem from algebra that a polynomial of degree k hasno more than k roots, and prove that P is 1-1 on some interval [M, ?).

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8. (10 pts) Suppose that f is twice continuously differentiable in aneighborhood of the point a, and f??(a) = 3. Compute

f(x) ? f(a) ? f?(a)(x ? a)lim 2 .x?a (x ? a)

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9. (10 pts) Suppose that p > 0, and the function f is equal to|x|p cos( x1 ) if x ?= 0, and f (0) = 0. Prove that f is differentiable at zeroif and only if p > 1.

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10. (10 pts) Prove that for every x > 0

1 + 2 ln x ? x .

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Extra Credit (10 pts). Suppose that f is twice continuously differ-entiable on [0,1], f(0) = f(1) = 0 and |f??(x)| ? A for every x ? (0,1).Prove that |f?(x)| ? A2 for every x ? [0,1].