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# Make up 1 - Rehman(aar638 Makeup Exam 1 sachse(56620 This...

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Rehman (aar638) – Makeup Exam 1 – sachse – (56620) 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 Use properties of integrals to determine the value of I = integraldisplay 5 0 f ( x ) dx when integraldisplay 7 0 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 0 f ( x ) dx = integraldisplay 5 0 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 0 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 0 f ( x ) g ( x ) dx = integraldisplay 2 0 1 dx = 2 , while parenleftBig integraldisplay 2 0 f ( x ) dx parenrightBigparenleftBig integraldisplay 2 0 g ( x ) dx parenrightBig = 2 × 2 = 4 . On the other hand, in III, integraldisplay 2 0 radicalbig f ( x ) dx = integraldisplay 2 0 1 dx = 2 ,

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Rehman (aar638) – Makeup Exam 1 – sachse – (56620) 2 while radicalBigg integraldisplay 2 0 f ( x ) dx = radicalBigg integraldisplay 2 0 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 anti-derivatives of f ? 1. 2. 3. 4. 5. correct 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 a horizontal and vertical translation be the
Rehman (aar638) – Makeup Exam 1 – sachse – (56620) 3 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 .

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Make up 1 - Rehman(aar638 Makeup Exam 1 sachse(56620 This...

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