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Unformatted text preview: Version 007 – Exam 1 – cheng – (58520) 1 This printout should have 17 questions. Multiplechoice 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 antiderivatives of f ? 1. 2. 3. 4. 5. 6. cor rect 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. In general, no horizontal translation of the graph of an antiderivative can be the Version 007 – Exam 1 – cheng – (58520) 2 graph of an antiderivative, nor can a hori zontal and vertical translation be the graph of an antiderivative. 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 antiderivatives of f , therefore, we have to look more carefully at the actual graphs. But calculus ensures that (i) an antiderivative of f will have a local extremum at the xintercepts of f . This eliminates two more figures since they contains graphs whose local extrema occur at points other than the xintercepts of f . (ii) An antiderivative of f is increasing on interval where the graph of f lies above the xaxis, and decreasing where the graph of f lies below the xaxis. Consequently, of the two remaining figures only consists entirely of graphs of antiderivatives of f . 002 10.0 points Rewrite the sum 6 n parenleftBig 2 + 5 n parenrightBig 2 + 6 n parenleftBig 2 + 10 n parenrightBig 2 + . . . + 6 n parenleftBig 2 + 5 n n parenrightBig 2 using sigma notation. 1. n summationdisplay i = 1 6 n parenleftBig 2 + 5 i n parenrightBig 2 correct 2. n summationdisplay i = 1 6 i n parenleftBig 2 + 5 i n parenrightBig 2 3. n summationdisplay i = 1 5 n parenleftBig 2 i + 6 i n parenrightBig 2 4. n summationdisplay i = 1 5 i n parenleftBig 2 + 6 i n parenrightBig 2 5. n summationdisplay i = 1 5 n parenleftBig 2 + 6 i n parenrightBig 2 6. n summationdisplay i = 1 6 n parenleftBig 2 i + 5 i n parenrightBig 2 Explanation: The terms are of the form 6 n parenleftBig 2 + 5 i n parenrightBig 2 , with i = 1 , 2 , . . . , n . Consequently in sigma notation the sum becomes n summationdisplay i = 1 6 n parenleftBig 2 + 5 i n parenrightBig 2 . 003 10.0 points Find an expression for the area of the region under the graph of f ( x ) = x 4 on the interval [5 , 9]....
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This note was uploaded on 12/11/2009 for the course M 408L taught by Professor Radin during the Fall '08 term at University of Texas.
 Fall '08
 RAdin
 Calculus

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