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Unformatted text preview: husain (aih243) Exam03Review Gilbert (56215) 1 This printout should have 25 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Which one of the following could be the graph of f ( x ) = log 2 (3 x ) when dashed lines indicates asymptotes? 1. 2 4 2 2 correct 2. 2 4 2 2 3. 2 4 2 2 4. 2 4 2 2 5. 2 4 2 2 6. 2 4 2 2 Explanation: Lets first review some properties of log 2 ( x ) and log 2 ( x ). Since log 2 (1) = 0, the graph of log 2 ( x ) has xintercept at x = 1. On the other hand, log 2 ( x ) is defined only on (0 , ) and lim x + log 2 ( x ) = , lim x log 2 ( x ) = , so x = 0 is a vertical asymptote. Thus the graph of log 2 ( x ) is husain (aih243) Exam03Review Gilbert (56215) 2 2 4 2 2 To get the graph of log 2 ( x ) we simply flip the one for log 2 ( x ) over both the xaxis and the yaxis, producing 2 4 2 2 To obtain the graph of y = log 2 (3 x ) from this last one all we have to do now is translate horizontally to the right by 3, pro ducing 2 4 2 2 keywords: LogFunc, LogFuncExam, 002 10.0 points If f is a continuous function such that integraldisplay x f ( t ) dt = 9 x x 2 + 7 , find the value of f (1). 1. f (1) = 57 64 2. f (1) = 27 32 correct 3. f (1) = 7 8 4. f (1) = 29 32 5. f (1) = 55 64 Explanation: By the Fundamental Theorem of Calculus, d dx parenleftBig integraldisplay x f ( t ) dt parenrightBig = f ( x ) . So by the Quotient Rule, f ( x ) = d dx parenleftBig 9 x x 2 + 7 parenrightBig = 63 9 x 2 ( x 2 + 7) 2 . In this case, f (1) = 27 32 . keywords: indefinite integral, Fundamental Theorem Calculus, FTC, function value, Quo tient Rule, rational function, 003 10.0 points Evaluate the definite integral I = integraldisplay / 2 (5 cos x 2 sin x ) dx . husain (aih243) Exam03Review Gilbert (56215) 3 1. I = 4 2. I = 6 3. I = 7 4. I = 3 correct 5. I = 5 Explanation: By the Fundamental Theorem of Calculus, I = bracketleftBig F ( x ) bracketrightBig / 2 = F ( 2 ) F (0) for any antiderivative F of f ( x ) = 5 cos x 2 sin x . Taking F ( x ) = 5 sin x + 2 cos x and using the fact that cos 0 = sin 2 = 1 , sin 0 = cos 2 = 0 , we thus see that I = 3 . 004 10.0 points Evaluate the definite integral I = integraldisplay 8 3  x 6  dx. 1. I = 15 2 2. I = 8 3. I = 15 2 4. I = 7 5. I = 13 2 6. I = 13 2 correct 7. I = 7 Explanation: Since  x 6  = braceleftbigg 6 x, x < 6, x 6 , x 6, we split the integral I into two parts I = integraldisplay 6 3 (6 x ) dx + integraldisplay 8 6 ( x 6) dx = I 1 + I 2 . Then I 1 = bracketleftBig 6 x 1 2 x 2 bracketrightBig 6 3 = 9 2 ....
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 Spring '06
 McAdam

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