HW03-solutions - wei (jw35975) HW03 kalahurka (55230) 1...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 / 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 / 4 2 sin x dx = bracketleftBig 2 cos x bracketrightBig / 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 ....
View Full Document

Page1 / 12

HW03-solutions - wei (jw35975) HW03 kalahurka (55230) 1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online