HW 10 - mandel (tgm245) – HW10 – Radin – (56470) 1...

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Unformatted text preview: mandel (tgm245) – HW10 – Radin – (56470) 1 This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Let f be the function defined by f ( x ) = 3 + 2 x 1 / 3 . Consider the following properties: A. derivative exists for all x ; B. has vertical tangent at x = 0 ; C. concave up on (0 , ∞ ) . Which does f have? 1. A and B only 2. B and C only 3. A and C only 4. C only 5. A only 6. None of them 7. B only correct 8. All of them Explanation: The graph of f is 2 4 − 2 − 4 2 4 6 On the other hand, after differentiation, f ′ ( x ) = 2 3 x 2 / 3 , f ′′ ( x ) = − 4 9 x 5 / 3 . Consequently, A. not have: ( f ′ ( x ) = (2 / 3) x − 2 / 3 , x negationslash = 0); B. has: (see graph); C. not have: ( f ′′ ( x ) < , x > 0). 002 10.0 points If f is increasing and its graph is concave up on (0 , 1), which of the following could be the graph of the derivative , f ′ , of f ? 1. 1 f ′ ( x ) cor- rect 2. f ′ ( x ) 1 3. 1 f ′ ( x ) mandel (tgm245) – HW10 – Radin – (56470) 2 4. f ′ ( x ) 1 Explanation: The function f increases when f ′ > 0 on (0 , 1), and its graph is concave up when f ′′ > 0. Thus on (0 , 1) the graph of f ′ lies above the x-axis and is increasing. Of the four graphs, only 1 f ′ ( x ) has these properties. 003 10.0 points When Sue uses first and second derivatives to analyze a particular continuous function y = f ( x ) she obtains the chart y y ′ y ′′ x < − 3 + − x = − 3 4 − 3 < x < − − x = 0 1 − 1 < x < 2 − + x = 2 − 1 DNE x > 2 + + Which of the following can she conclude from her chart? A. f is concave up on ( −∞ , 0) . B. f has a point of inflection at x = 0. C. f is concave up on (0 , 2) . 1. all of them 2. C only 3. A and B only 4. B and C only correct 5. A and C only 6. B only 7. none of them 8. A only Explanation: The graph of f must look like 2 − 2 − 4 2 4 Consequently, A. False. B. True. C. True. 004 10.0 points The figure below shows the graphs of three functions: mandel (tgm245) – HW10 – Radin – (56470) 3 One is the graph of a function f , one is its derivative f ′ , and one is its second derivative f ′′ . Identify which graph goes with which function. 1. f : f ′ : f ′′ : correct 2. f : f ′ : f ′′ : 3. f : f ′ : f ′′ : 4. f : f ′ : f ′′ : 5. f : f ′ : f ′′ : 6. f : f ′ : f ′′ : Explanation: Calculus tells us that f (i) has horizontal tangent at ( x , f ( x )) when f ′ crosses the x-axis, (ii) is increasing when f ′ > 0, and (iii) is decreasing when f ′ < 0, (iv) has a local max at x when f ′ ( x ) = 0 and f ′′ ( x ) < 0, (v) has a local min at x when f ′ ( x ) = 0 and f ′′ ( x ) > 0, (vi) is concave up when f ′′ > 0, (v) and concave down when f ′′ < 0....
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This note was uploaded on 11/07/2010 for the course M 408k taught by Professor Schultz during the Spring '08 term at University of Texas.

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HW 10 - mandel (tgm245) – HW10 – Radin – (56470) 1...

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