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Unformatted text preview: Rehman (aar638) Makeup Exam 1 sachse (56620) 1 This printout should have 18 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. 001 10.0 points Use properties of integrals to determine the value of I = integraldisplay 5 f ( x ) dx when integraldisplay 7 f ( x ) dx = 11 , integraldisplay 7 5 f ( x ) dx = 8 . 1. I = 1 2. I = 1 3. I = 2 4. I = 3 correct 5. I = 0 Explanation: Since integraldisplay 7 f ( x ) dx = integraldisplay 5 f ( x ) dx + integraldisplay 7 5 f ( x ) dx , we see that I = 11 8 = 3 . keywords: definite integral, conceptual, prop erties of integrals, 002 10.0 points If f and g are continuous functions such that f ( x ) 0 for all x , which of the following must be true? I . integraldisplay b a f ( x ) g ( x ) dx = parenleftBig integraldisplay b a f ( x ) dx parenrightBigparenleftBig integraldisplay b a g ( x ) dx parenrightBig II . integraldisplay b a braceleftBig f ( x ) + g ( x ) bracerightBig dx = integraldisplay b a f ( x ) dx + integraldisplay b a g ( x ) dx III . integraldisplay b a radicalbig f ( x ) dx = radicalBigg integraldisplay b a f ( x ) dx 1. III only 2. I and II only 3. II and III only 4. II only correct 5. I only Explanation: Property II is a special case of the linearity property of integrals, i.e. integral of the sum = sum of the integrals . But there is no product rule or square root rule for integrals. For example, we know that integraldisplay 2 f ( x ) dx = 2 when f ( x ) = 1, because the value of the inte gral is the area of a rectangle of height 1 and base length 2. So when f ( x ) = g ( x ) = 1 in I, integraldisplay 2 f ( x ) g ( x ) dx = integraldisplay 2 1 dx = 2 , while parenleftBig integraldisplay 2 f ( x ) dx parenrightBigparenleftBig integraldisplay 2 g ( x ) dx parenrightBig = 2 2 = 4 . On the other hand, in III, integraldisplay 2 radicalbig f ( x ) dx = integraldisplay 2 1 dx = 2 , Rehman (aar638) Makeup Exam 1 sachse (56620) 2 while radicalBigg integraldisplay 2 f ( x ) dx = radicalBigg integraldisplay 2 1 dx = 2 . Consequently, only II is true . 003 10.0 points If the graph of f is which one of the following contains only graphs of antiderivatives of f ? 1. 2. 3. 4. 5. correct 6. Explanation: If F 1 and F 2 are antiderivatives of f then F 1 ( x ) F 2 ( x ) = constant independently of x ; this means that for any two antiderivatives 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 antiderivative of f will be the graph of an antiderivative of f , nor can a horizontal and vertical translation be the Rehman (aar638) Makeup Exam 1 sachse (56620) 3 graph of an antiderivative. This rules out two sets of graphs....
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This note was uploaded on 02/01/2011 for the course MATH 408L taught by Professor Gogolev during the Fall '09 term at University of Texas at Austin.
 Fall '09
 GOGOLEV
 Calculus

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