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30 Jan 2018

[6] 4. (a) If a function f has a derivative at a number a in its domain, prove that f is continuous at a. Your proof must demonstrate an understanding of what it means for a function to have a derivative, and what it means for it to be continuous. for r +0 In our son={** (2) set to move to 10-2 [4] (b) If f(x) = { prove that f'(0) = 0. for 1 = 0

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Nelly Stracke
Nelly StrackeLv2
1 Feb 2018
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Related questions

A fixed point of a function f is a number c in its domain such that . (The function doesn’t move c; it stays fixed.)

(a) Sketch the graph of a continuous function with domain whose range also lies in. Locate a fixed point of f.

(b) Try to draw the graph of a continuous function with domain and range in that does not have a fixed point. What is the obstacle?

(c) Use the Intermediate Value Theorem to prove that any continuous function with the domain and range in must have a fixed point.

[8] 9. Either prove the statement is true or show that it is false. (a) sin (sin(2)) = 27. Solution: It is FALSE. The range of sin-' r is (-1,1], hence sin (sin(27)) = sin (0) = 0 (b) If f is continuous at a, then f is continuous at a. Solution: It is TRUE. Let g(x) = \f()]. If lim f(x) = f(a), then we have since the absolute value function is continuous lim g(x) = lim f() = lim f(x) = f(a) = g(a) So, g = \f is continuous at a. (c) If [g] is continuous at a, then g is continuous at a. -1 if r <0 Solution: It is FALSE. Let g(x) = 3 · Then, g = 1, so g is continuous at 0 i if >0 by the Continuity Theorems, but lim g(t) = -1 + lim g(1) 240- 1- 0+ So, g(x) is not continuous at a. (d) If f is continuous at a, then f is differentiable at a. Solution: It is FALSE. We know that f(1) = 2 is continuous at 0, but is not differentiable at 0.

10. Exactly how many of the following statements are always true? (i) If f is continuous at 0 ar :) dx > 0, then f is differentiable at 0. (ii) The definite integral of a continuous function h over [a, b] is a function H such that / h(x) dx = H(b) – Ha). (iii) A continuous function g has a local minimum at a number c if and only if c is a critical number of g. (iv) If a is a critical number of a polynomial, then the 2nd derivative test is always preferable over the 1st derivative test when checking for extrema at a. (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

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