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Unformatted text preview: silva (jrs4378) HW 10 gualdani (56455) 1 This printout should have 23 questions. Multiplechoice 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 negationslash = 0 ; B. concave up on (0 , ) ; C. has vertical tangent at x = 0 . Which does f have? 1. B only 2. C only 3. A only 4. A and C only correct 5. None of them 6. A and B only 7. All of them 8. B and C only 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. has: ( f ( x ) = (2 / 3) x 2 / 3 , x negationslash = 0); B. not have: ( f ( x ) &lt; , x &gt; 0); C. has: (see graph). 002 10.0 points If f is increasing and its graph is concave down on (0 , 1), which of the following could be the graph of the derivative , f , of f ? 1. 1 f ( x ) 2. f ( x ) 1 cor rect 3. f ( x ) 1 silva (jrs4378) HW 10 gualdani (56455) 2 4. 1 f ( x ) Explanation: The function f increases when f &gt; 0 on (0 , 1), and its graph is concave down when f &lt; 0. Thus on (0 , 1) the graph of f lies above the xaxis and is decreasing. Of the four graphs, only f ( x ) 1 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 &lt; 3 + x = 3 4 3 &lt; x &lt; x = 0 1 1 &lt; x &lt; 2 + x = 2 1 DNE x &gt; 2 + + Which of the following can she conclude from her chart? A. f is concave up on ( , 0) . B. f is concave down on (0 , ) . C. f has a point of inflection at x = 2. 1. none of them correct 2. A and C only 3. B and C only 4. all of them 5. A and B only 6. B only 7. A only 8. C only Explanation: The graph of f must look like 2 2 4 2 4 Consequently, A. False. B. False. C. False. 004 10.0 points The figure below shows the graphs of three functions: silva (jrs4378) HW 10 gualdani (56455) 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 xaxis, (ii) is increasing when f &gt; 0, and (iii) is decreasing when f &lt; 0, (iv) has a local max at x when f ( x ) = 0 and f ( x ) &lt; 0, (v) has a local min at x when f ( x ) = 0 and f ( x ) &gt; 0, (vi) is concave up when f &gt; 0, (v) and concave down when f &lt; 0....
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 Spring '10
 Gualdani
 Derivative, lim

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