Exam03-solutions - husain (aih243) Exam03Review Gilbert...

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Unformatted text preview: husain (aih243) Exam03Review Gilbert (56215) 1 This print-out should have 25 questions. Multiple-choice 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 x-intercept 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 x-axis and the y-axis, 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 anti-derivative 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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Exam03-solutions - husain (aih243) Exam03Review Gilbert...

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