HW03-solutions

# HW03-solutions - wei(jw35975 – HW03 – kalahurka...

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Unformatted text preview: wei (jw35975) – HW03 – kalahurka – (55230) 1 This print-out should have 25 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Evaluate the integral I = integraldisplay π/ 4 sin 2 x cos x dx . 1. I = 2 − √ 2 correct 2. I = 1 2 (2 − √ 2) 3. I = 2(2 − √ 3) 4. I = 1 2 5. I = 1 6. I = 2 Explanation: Since sin 2 x = 2 sin x cos x , the integrand can be rewritten as sin 2 x cos x = 2 sin x . But then I = integraldisplay π/ 4 2 sin x dx = bracketleftBig − 2 cos x bracketrightBig π/ 4 . Consequently, I = 2 − √ 2 . 002 10.0 points A car heads slowly north from Austin on IH 35. Its velocity t hours after leaving Austin is given (in miles per hour) by v ( t ) = 8 + 2 t − t 2 . How many miles will the car have covered during the first 2 hours of driving? 1. distance = 52 3 miles correct 2. distance = 61 3 miles 3. distance = 64 3 miles 4. distance = 58 3 miles 5. distance = 55 3 miles Explanation: The total distance travelled during the first T hours of driving is given by I = integraldisplay T | v ( t ) | dt, where v = v ( t ) is the velocity of the car. When v ( t ) = 8 + 2 t − t 2 , T = 2 , therefore, I = integraldisplay 2 | 8 + 2 t − t 2 | dt . To eliminate the absolute value in the inte- grand we need to look at the graph of v ( t ). This graph is a parabola opening downwards and its t-intercepts are the solutions of v ( t ) = 8 + 2 t − t 2 = 0 , i.e. , at t = − 2 and t = 4. But 2 < 4, so | v ( t ) | = v ( t ) on [0 , 2]. Thus I = integraldisplay 2 (8 + 2 t − t 2 ) dt = bracketleftBig 8 t + t 2 − 1 3 t 3 bracketrightBig 2 . wei (jw35975) – HW03 – kalahurka – (55230) 2 Consequently, distance = 52 3 miles . keywords: word problem, definite integral, polynomial, distance travelled, velocity, 003 10.0 points If on [0 , 8] 2 4 6 8 2 4 6 8 is the graph of an anti-derivative, F , of a function f , use this graph to determine the value of the definite integral I = integraldisplay 8 | f ( x ) | dx . 1. I = 5 2. I = 8 3. I = 7 4. I = 6 5. I = 9 correct Explanation: Since F ′ ( x ) = f ( x ), the slope of the graph of F determines the sign of f . Thus f ( x ) > on (0 , 4), while f ( x ) < 0 on (4 , 8). Conse- quently, integraldisplay 8 | f ( x ) | dx = integraldisplay 4 f ( x ) dx − integraldisplay 8 4 f ( x ) dx . On the other hand, by the Fundamental The- orem of Calculus, integraldisplay 4 f ( x ) dx = F (4) − F (0) , while integraldisplay 8 4 f ( x ) dx = F (8) − F (4) . After reading the values of F from the graph, therefore, we see that I = 2 F (4) − F (0) − F (8) = 9 . keywords: graph analysis, fundamental theo- rem 004 10.0 points Determine the integral I = integraldisplay 2 − 5 x √ x dx ....
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HW03-solutions - wei(jw35975 – HW03 – kalahurka...

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