HW 10 - mandel(tgm245 HW10 Radin(56470 This print-out...

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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: 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 ? A. derivative exists for all x ; B. has vertical tangent at x = 0 ; C. concave up on (0 , ) . Consequently, A. not have: ( f ( x ) = (2 / 3) x 2 / 3 , x negationslash = 0); B. has: (see graph); C. not have: ( f ′′ ( x ) < 0 , x > 0).
Which does f have?
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 0 Consequently, 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 4 004 10.0 points
Which of the following can she conclude from her chart? 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 0 , f ( x 0 )) 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 0 when f ( x 0 ) = 0 and f ′′ ( x 0 ) < 0, (v) has a local min at x 0 when f ( x 0 ) = 0 and f ′′ ( x 0 ) > 0, (vi) is concave up when f ′′ > 0, (v) and concave down when f ′′ < 0. Inspection of the graphs thus shows that f : f : f ′′ : . 005 10.0 points Find all intervals on which f ( x ) = x 2 ( x + 2) 3 is decreasing. 1. ( −∞ , 2) , ( 2 , 0] , [4 , ) correct 2. [ 4 , 0] 3. ( −∞ , 4] , [0 , ) 4. [0 , 4] 5. ( −∞ , 0] , [4 , ) 6. ( −∞ , 4] , [0 , 2) , (2 , ) 7. ( 2 , 2) Explanation: By the Quotient Rule, f ( x ) = 2 x ( x + 2) 3 3 x 2 ( x + 2) 2 ( x + 2) 6 = 2 x ( x + 2) 3 x 2 ( x + 2) 4 = x (4 x ) ( x + 2) 4 . Now f will be decreasing on an interval [ a, b ] (resp. [ a, b )) when f ( x ) < 0 on the interval ( a, b ) (resp. [ a, b ) if f ( b ) is not defined). In this case the inequality f ( x ) < 0 will hold when x (4 x ) < 0 , x negationslash = 2 .
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