xm 3 - Version 085 – EXAM 3 – Fouli – (58320) 1 This...

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Unformatted text preview: Version 085 – EXAM 3 – Fouli – (58320) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find the most general antiderivative, F , of the function f ( x ) = 12 x 2 − 16 x + 6 . 1. F ( x ) = 4 x 3 − 8 x 2 + 6 x 2. F ( x ) = 4 x 3 − 8 x 2 + 6 x + C correct 3. F ( x ) = 4 x 3 + 8 x 2 + 6 x + C 4. F ( x ) = 4 x 3 + 8 x 2 + 6 x 5. F ( x ) = 12 x 3 − 16 x 2 + 6 x + C Explanation: Since d dx x r = rx r − 1 , the most general anti-derivative of f is the function F ( x ) = 12 parenleftbigg x 3 3 parenrightbigg − 16 parenleftbigg x 2 2 parenrightbigg + 6 x + C with C an arbitrary constant. Consequently, F ( x ) = 4 x 3 − 8 x 2 + 6 x + C . 002 10.0 points If the graph of f is which one of the following contains only graphs of anti-derivatives of f ? 1. cor- rect 2. 3. 4. Version 085 – EXAM 3 – Fouli – (58320) 2 5. 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. In general, no horizontal translation of the graph of an anti-derivative can be the 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 . keywords: antiderivative, graphical, graph, geometric interpretation /* If you use any of these, fix the comment symbols. 003 10.0 points Find the value of f (0) when f ′ ( t ) = cos 2 t , f parenleftBig π 4 parenrightBig = 3 . 1. f (0) = 5 2 correct 2. f (0) = 3 3. f (0) = 4 4. f (0) = 7 2 5. f (0) = 2 Explanation: Since d dx sin mt = m cos mt , for all m negationslash = 0, we see that f ( t ) = 1 2 sin 2 t + C Version 085 – EXAM 3 – Fouli – (58320) 3 where the arbitrary constant C is determined by the condition f ( π/ 4) = 3. But sin2 t vextendsingle vextendsingle vextendsingle t = π/ 4 = sin π 2 = 1 ....
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This note was uploaded on 09/09/2009 for the course M 408L taught by Professor Gilbert during the Spring '09 term at University of Texas-Tyler.

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xm 3 - Version 085 – EXAM 3 – Fouli – (58320) 1 This...

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