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AnswersCalcHW3 - Version 010 – Homework 3 – Odell...

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Unformatted text preview: Version 010 – Homework 3 – Odell – (58340) 1 This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points The graph of f is shown in the figure 2 4 6 8 2 4 6 If F is an anti-derivative of f and integraldisplay 8 2 f ( x ) dx = 49 2 , find the value of F (8)- F (1). 1. F (8)- F (1) = 28 2. F (8)- F (1) = 119 4 3. F (8)- F (1) = 217 8 correct 4. F (8)- F (1) = 245 8 5. F (8)- F (1) = 231 8 Explanation: We already know that the area under the graph on the interval 2 ≤ x ≤ 8 is equal to 49 2 , alternatively, by the Fundamental Theorem of Calculus we can say that F (8)- F (2) = 49 2 . On the other hand, integraldisplay 8 1 f ( x ) dx = integraldisplay 2 1 f ( x ) dx + integraldisplay 8 2 f ( x ) dx. Thus we need to find integraldisplay 2 1 f ( x ) dx = F (2)- F (1) . Now integraldisplay 2 1 f ( x ) dx = integraldisplay 2 1 7 4 x dx = 7 8 bracketleftBig x 2 bracketrightBig 2 1 = 21 8 . Consequently, F (8)- F (1) = 49 2 + 21 8 = 217 8 . keywords: velocity, distance, graph analysis, fundamental theorem 002 10.0 points Calculate the indefinite integral I = integraldisplay (4- √ x )(5 + √ x ) dx . 1. I = 4 x- 2 3 x √ x + 1 2 x 2 + C 2. I = 20 x + √ x- 1 2 x 2 + C 3. I = 20 x- 2 3 x √ x- 1 2 x 2 + C correct 4. I = 4 x + √ x- 1 2 x 2 + C 5. I = 20 x- 2 3 x √ x + 1 2 x 2 + C 6. I = 4 x + 2 3 x √ x- 1 2 x 2 + C Explanation: After expansion (4- √ x )(5 + √ x ) = 20- √ x- x . Version 010 – Homework 3 – Odell – (58340) 2 Thus I = integraldisplay ( 20- √ x- x ) dx = 20 x- 2 3 x √ x- 1 2 x 2 + C . Consequently, I = 20 x- 2 3 x √ x- 1 2 x 2 + C . 003 10.0 points Evaluate the definite integral I = integraldisplay π 3 3 sin 2 x- 2 cos 2 x cos x dx . 1. I = 6 + √ 2 2. I = 6 + √ 3 3. I = 6- √ 3 4. I = 3- √ 2 5. I = 3 + √ 3 6. I = 3- √ 3 correct Explanation: Since sin 2 x = 2 sin x cos x , the integrand can be rewritten as 6 sin x cos x- 2 cos 2 x cos x = 2(3 sin x- cos x ) . Thus I = 2 integraldisplay π 3 (3 sin x- cos x ) dx = 2 bracketleftBig- 3 cos x- sin x bracketrightBig π 3 = 2 parenleftBigg- 3 2- √ 3 2 parenrightBigg + 6 . Consequently, I = 3- √ 3 . 004 10.0 points Evaluate the integral I = integraldisplay 2 d dx (3 + x 2 ) 1 / 2 dx. 1. I = √ 7 + √ 3 2. I = √ 7 3. I = √ 3- √ 7 4. I = √ 3 5. I = √ 7- √ 3 correct Explanation: As an indefinite integral, integraldisplay d dx (3 + x 2 ) 1 / 2 dx = (3 + x 2 ) 1 / 2 + C where C is an arbitrary constant. Thus integraldisplay 2 d dx (3 + x 2 ) 1 / 2 dx = bracketleftBig (3 + x 2 ) 1 / 2 bracketrightBig 2 . Consequently, I = √ 7- √ 3 . 005 10.0 points Determine the indefinite integral I = integraldisplay 4- cos 2 θ cos 2 θ dθ ....
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AnswersCalcHW3 - Version 010 – Homework 3 – Odell...

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