Review 3-solutions-1

# Review 3-solutions-1 - white (taw933) Review 3 ben-zvi...

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Unformatted text preview: white (taw933) Review 3 ben-zvi (55600) 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. correct 4. 5. 6. 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 white (taw933) Review 3 ben-zvi (55600) 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 ) = 2 cos x + 2 sec 2 x and f ( 4 ) = 4. 1. f ( x ) = 7- 2 tan x- 2 cos x 2. f ( x ) = 2 tan x- 2 cos x + 1 3. f ( x ) = 2 tan x + 2 sin x + 1 correct 4. f ( x ) = 2 tan x- 2 sin x + 3 5. f ( x ) = 5- 2 tan x + 2 sin x Explanation: The most general anti-derivative of f ( x ) = 2 cos x + 2 sec 2 x is f ( x ) = 2 sin x + 2 tan x + C with C an arbitrary constant. But if f parenleftBig 4 parenrightBig = 4, then f parenleftBig 4 parenrightBig = 1 + 2 + C = 4 , so C = 1 . Consequently, f ( x ) = 2 tan x + 2 sin x + 1 . 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- 12 (in units of feet and seconds). If the initial velocity of the particle is 32 feet/sec, at what time t during the interval t 6 does the particle change direction? 1. t = 5 seconds 2. t = 2 seconds 3. t = 6 seconds 4. none of the other answers 5. t = 3 seconds white (taw933) Review 3 ben-zvi (55600) 3 6. t = 4 seconds correct Explanation: The acceleration at time t is given by a ( t ) = dv dt = 2 t- 12 ....
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## Review 3-solutions-1 - white (taw933) Review 3 ben-zvi...

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