This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Panjwani, Sameer – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: James Rath 1 This printout should have 18 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Below is the graph of a function f . 2 4 6 8 10 12 2 4 6 8 2 4 6 Estimate the definite integral I = Z 12 f ( x ) dx with six equal subintervals using left end points. 1. I ≈ 14 correct 2. I ≈ 6 3. I ≈ 10 4. I ≈ 12 5. I ≈ 8 Explanation: Since [0 , 12] is subdivided into six equal subintervals, each of these will have length 2 and the six corresponding rectangles are shown as the shaded areas in 2 4 6 8 10 12 2 4 6 8 2 4 6 The heights of the rectangles are left endpoint sample values of f that can be read off from the graph. Thus, with left endpoints, I ≈ 2(4 + 1 4 1 + 3 + 4) = 14 . keywords: graph, Riemann sum, left end points 002 (part 1 of 1) 10 points If F ( x ) = Z x 5 e 8 sin 2 θ dθ , find the value of F ( π/ 4). 1. F ( π/ 4) = 4 e 4 2. F ( π/ 4) = 4 e 5 3. F ( π/ 4) = 4 e 8 4. F ( π/ 4) = 5 e 4 correct 5. F ( π/ 4) = 5 e 8 Explanation: By the Fundamental theorem of calculus, F ( x ) = 5 e 8 sin 2 x . At x = π/ 4, therefore, F ( π/ 4) = 5 e 4 Panjwani, Sameer – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: James Rath 2 since sin( π 4 ) = 1 √ 2 . keywords: integral, FTC 003 (part 1 of 1) 10 points Determine F ( x ) when F ( x ) = Z √ x 5 8 sin t t dt. 1. F ( x ) = 8 cos x x 2. F ( x ) = 8 cos x √ x 3. F ( x ) = 4 sin( √ x ) x correct 4. F ( x ) = 8 cos( √ x ) √ x 5. F ( x ) = 4 sin( √ x ) √ x 6. F ( x ) = 8 sin x √ x 7. F ( x ) = 4 sin x x 8. F ( x ) = 4 cos( √ x ) x Explanation: By the Fundamental Theorem of Calculus and the Chain Rule, d dx ‡ Z g ( x ) a f ( t ) dt · = f ( g ( x )) g ( x ) . When F ( x ) = Z √ x 5 8 sin t t dt, therefore, F ( x ) = 8 sin( √ x ) √ x ‡ d dx √ x · . Consequently, F ( x ) = 4 sin( √ x ) x , since d dx √ x = 1 2 √ x . keywords: Stewart5e, FTC, Chain Rule 004 (part 1 of 1) 10 points If w ( t ) is the rate of growth of Mira’s weight (in pounds per year), what does the definite integral I = Z 8 3 w ( t ) dt represent? 1. decrease in Mira’s weight from age 3 to 8 2. average of Mira’s weight from age 3 to 8 3. Mira’s weight at age 8 4. increase in Mira’s weight from age 3 to 8 correct 5. Mira’s weight at age 3 Explanation: By the Fundamental theorem of Calculus, Z b a w ( x ) dx = w ( b ) w ( a ) , in other words, the value of the integral is the net change, w ( b ) w ( a ), in w over the interval [ a, b ]....
View
Full
Document
This note was uploaded on 03/18/2008 for the course M 408L taught by Professor Radin during the Fall '08 term at University of Texas at Austin.
 Fall '08
 RAdin
 Calculus

Click to edit the document details