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Unformatted text preview: Version 063 Exam 3 gualdani (56410) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points If the graph of f is which one of the following contains only graphs of anti-derivatives of f ? 1. 2. 3. 4. 5. 6. correct Explanation: If F 1 and F 2 are anti-derivatives of f then F 1 ( x ) F 2 ( x ) = constant independently of x ; this means that for any two anti-derivatives of f the graph of one is just a vertical translation of the graph of the other. But no horizontal translation of the graph of an anti-derivative of f will be the graph of an anti-derivative of f , nor can Version 063 Exam 3 gualdani (56410) 2 a horizontal and vertical translation be the graph of an anti-derivative. This rules out two sets of graphs. Now in each of the the remaining four fig- ures the dotted and dashed graphs consist of vertical translations of the graph whose line- style is a continuous line. To decide which of these figures consists of anti-derivatives of f , therefore, we have to look more carefully at the actual graphs. But calculus ensures that (i) an anti-derivative of f will have a local extremum at the x-intercepts of f . This eliminates two more figures since they contains graphs whose local extrema occur at points other than the x-intercepts of f . (ii) An anti-derivative of f is increasing on interval where the graph of f lies above the x-axis, and decreasing where the graph of f lies below the x-axis. Consequently, of the two remaining figures only consists entirely of graphs of anti-derivatives of f . keywords: antiderivative, graphical, graph, geometric interpretation 002 10.0 points Find f ( x ) on ( 2 , 2 ) when f ( x ) = 3 2 sin x + 2 sec 2 x and f ( 4 ) = 3. 1. f ( x ) = 2 2 tan x + 3 2 sin x 2. f ( x ) = 2 tan x 3 2 cos x + 4 correct 3. f ( x ) = 8 2 tan x 3 2 cos x 4. f ( x ) = 2 tan x + 3 2 sin x + 4 5. f ( x ) = 2 tan x + 3 2 cos x 2 Explanation: The most general anti-derivative of f ( x ) = 3 2 sin x + 2 sec 2 x is f ( x ) = 3 2 cos x + 2 tan x + C with C an arbitrary constant. But if f parenleftBig 4 parenrightBig = 3, then f parenleftBig 4 parenrightBig = 3 + 2 + C = 3 , so C = 4 . Consequently, f ( x ) = 2 tan x 3 2 cos x + 4 . 003 10.0 points A particle moves along a straight line so that its acceleration at any given time t is a ( t ) = 2 t 10 (in units of feet and seconds)....
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- Fall '09