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Unformatted text preview: white (taw933) – HW10 – benzvi – (55600) 1 This printout should have 19 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 ) = 5 − x 2 / 3 . Consider the following properties: A. derivative exists for all x ; B. has local minimum at x = 0 ; C. concave down on ( −∞ , 0) ∪ (0 , ∞ ) ; Which does f have? 1. B only 2. A and B only 3. A only 4. C only 5. B and C only 6. All of them 7. None of them correct 8. A and C only Explanation: The graph of f is 2 4 − 2 − 4 2 4 On the other hand, after differentiation, f ′ ( x ) = − 2 3 x 1 / 3 , f ′′ ( x ) = 2 9 x 4 / 3 . Consequently, A. not have: ( f ′ ( x ) = − (2 / 3) x − 1 / 3 , x negationslash = 0; B. not have: (see graph); C. not have: ( f ′′ ( x ) > , x negationslash = 0). 002 10.0 points Use the graph a b c of the derivative of f to locate the critical points x at which f has a local maximum? 1. x = c , a 2. x = a 3. x = c 4. x = a , b , c 5. none of a , b , c 6. x = b correct 7. x = b , c 8. x = a , b Explanation: Since the graph of f ′ ( x ) has no ‘holes’, the only critical points of f occur at the x intercepts of the graph of f ′ , i.e. , at x = a, b, and c . Now by the first derivative test, f will have white (taw933) – HW10 – benzvi – (55600) 2 (i) a local maximum at x if f ′ ( x ) changes from positive to negative as x passes through x ; (ii) a local minimum at x if f ′ ( x ) changes from negative to positive as x passes through x ; (iii) neither a local maximum nor a local minimum at x if f ′ ( x ) does not change sign as x passes through x . Consequently, by looking at the sign of f ′ ( x ) near each of x = a, b, and c we see that f has a local maximum only at x = b . 003 10.0 points If f is decreasing and its graph is concave down on (0 , 1), which of the following could be the graph of the derivative , f ′ , of f ? 1. f ′ ( x ) 1 cor rect 2. 1 f ′ ( x ) 3. 1 f ′ ( x ) 4. f ′ ( x ) 1 Explanation: The function f decreases when f ′ < 0 on (0 , 1), and its graph is concave down when f ′′ < 0. Thus on (0 , 1) the graph of f ′ lies below the xaxis and is decreasing. Of the four graphs, only 1 f ′ ( x ) has these properties. 004 10.0 points Ted makes a chart to help him analyze the continuous function y = f ( x ): white (taw933) – HW10 – benzvi – (55600) 3 y y ′ y ′′ x < − 1 + − x = − 1 2 − 1 < x < − − x = 0 1 − 1 < x < 2 − + x = 2 − 2 DNE x > 2 + + Consider the following statements: A. f has a point of inflection at x = 0. B. f has a local minimum at x = 2. C. f has a local minimum at x = − 1 ....
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 Spring '08
 schultz
 Derivative, Differential Calculus, lim, Mathematical analysis

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