EXAM1 - Version 007 Exam 1 cheng(58520 This print-out...

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Version 007 – Exam 1 – cheng – (58520) 1 This print-out should have 17 questions. Multiple-choice 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 anti-derivatives of f ? 1. 2. 3. 4. 5. 6. cor- rect 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. In general, no horizontal translation of the graph of an anti-derivative can be the
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Version 007 – Exam 1 – cheng – (58520) 2 graph of an anti-derivative, nor can a hori- zontal and vertical translation be the 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 . This eliminates two more figures since they contains graphs whose local extrema occur at points other than the x -intercepts of f . (ii) An anti-derivative of f is increasing on interval where the graph of f lies above the x -axis, and decreasing where the graph of f lies below the x -axis. Consequently, of the two remaining figures only consists entirely of graphs of anti-derivatives 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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