Math for Econ II Homework 8 - (c Are the answers to parts(a...

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Unformatted text preview: (c) Are the answers to parts (a) and (b) the same? Explain. ins5-2h22-28 2 5 2 In Problems ??, find the area of the regions between the curve f (x) Math for Econ II, Written Assignment 8 (28 points) and the horizontal axis 1 5-2h22 5-2h23 5-2h24 5-2h25 5-2h26 5-2h27 5-2h28 5-2h29 Due Friday, November 7 x 22. Under y = 6x3 2 for 5 x 10. Jankowski, Fall 2014 1 23. Under the curve y = cos t for 0 t /2. Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work, 2 5-2h32fig 24. you will = ln be given x 4. Under y not x for 1 2 4 6 8 10 any credit. 25. Under y = 2 cos(t/10) for 1 t 2. Figure 5.39: Graph consists of a semicircle and • Write your name x for 0 x 2. 26. Under the curve y = cos and recitation section number. line segments 27. Under the curve y =homework above the x-axis. 7 x2 and if you have multiple pages! • Staple your 28. Above the curve y = x4 8 and below the x-axis. 1. (2 pts to find the each) 29. Use Figure ??total; 1 ptvalues ofUse the figure below to find the values of 5-2h33 33. (a) Graph f (x) = x(x + 2)(x 1). Rb Rb Rc f (b) f (x) dx (a) (b) Find the total area between the graph and the x-axis (a)(x) dx(x) dx f R a Rc Rbc c between x = 2 and x = 1. c (c) f R1 a (b)(x)adx (x)| dx (d) a |f (x)| dx |f (c) Find 2 f (x) dx and interpret it in terms of areas. f (x) 5-2h34 Area = 13 5-2h35 a c b 5-2h29fig x 5-2h36 ! Area = 2 34. Compute the definite integral the result in terms of areas. 0 36. Estimate R1 0 e x2 cos R2 35. Without computation, decide if tive or negative. [Hint: Sketch e (a) Figure 5.36 R4 0 x x dx and interpret e x sin x dx is posisin x.] dx using n = 5 rectangles to form a Left-hand sum (b) Right-hand sum 5.4 THEOREMS ABOUT DEFINITE INTEGRALS 311 5-2h37 37. (a) On a sketch of y the ins5-4h47-48 In Problems ??, evaluate following = ln x, represent the left Riemann the expression, R 2 possible, or if = 2 approximating 1 2 2. (4in increasing order, from least to greatest. arrange whatfollowing quantities in increasingln x dx. that least to greatest. pts) Using the graph of f in Figure, the sum with n information is needed, order, Write from quantities2 R say additional given R R2 R4 R2R R2 R3 R2 out the terms in the sum, but do not evaluate it. R 1 f1 f (x)dx (a) i) (x)dx dx ii) (b) f (x)2dx iii) 2 f (x) f (x) dx g(x) dx 2 f (x) dx v) represent thedx Riemann sum iv) = 12. f (x) right vi) The number 0 0 f (x) dx (ii) 1 1 f (x) dx (i) 4 0 0 1 (b) On another sketch, R2 (c) 0 The total shaded area. R2 R3 (iii) f (x) dx (iv) f (x) dx Z 4 with n = 2 approximating 1 ln x dx. Write out the Z 4 0 2 R2 (x) terms in the sum, but do not evaluate it. 5-4h48 48. vii) (x) dx 20 viii)fThe 0 5-4h47 47. g(x) dx g( x) dx (v) f The number (vi) The numbernumber 10 R 5-4h43 43. Using the graph of f in Figure ??, arrange 5-2h30 30. Given 0 f (x)dx = 4 and Figure ??, estimate: (vii) 1 2 (viii) The number 20 The number x 10 5-2h30fig 5-4h43fig 2 2 1 10 f (x) 2 2 3 x Figure 5.37 10 (c) Which sum is an overestimate? Which sum is an un4 derestimate? 0 5-2h38 38. (a) Draw ins5-4h49-52 In Problems ??, ! the rectangles that give the left-hand sum apR evaluate 0 expression if possible, proximation to the sin x dx with n = 2. or say R7 R0 what extra information is needed, given dx. (x) dx = 25. (b) Repeat part (a) for sin x 0 f (c) From your answers to parts (a) and (b), what is Z 7 the value of the left-hand 3.5 approximation to Z sum R 5-4h50 50. sin f (x) dxx dx with n = 4? f (x) dx R0 Figure 5.71 5-4h49 49. 31. (a) Using Figure ??, find 3 f (x) dx. 0 0 R6 (b) R the area of the shaded region is A, estimate If 5-4h44 44. (a) Using Figures ?? and ??, find the average value on 5-2h39 39. (a) 4 R 1 4 computer7to R 4 (x + 1) dx. Z 5 R Use a calculator or Either findfind f0(x)2 dx, or show there is not Z 3. (3 f (x)of pts) dx. Suppose f is even, 2 f (x) dx = 3, and 2 f (x) dx = 5. 0 x 2 1 3 5-4h51 51. Represent value 52. R4 f (x + 2) dxthis5-4h52 as the area under a curve. (f (x) + 2) dx R6 2 (i) enough information to find 1 f (x) dx. f (x) (ii) g(x) (iii) f (x)·g(x) (b) Estimate 0 (x + 1) dx using a left-hand sum with 2 0 1 f (x) n = 3. Represent this sum graphically on a sketch (b) Is the following statement true? Explain your an4 4. (5 pts) Evaluate the following integrals:x of f (x) = x2 + 1. Is this sum an overestimate or swer. 4 3 2 1 2 3 55-4h53 53. 1(a) Sketch a graph of f (x) = sin(x2 ) and mark on it R11 R underestimatedx the9, find R 1found in part (a)? dx. true value 5-2h31fig (a) (3 pts) Average(g) = 2g(x)) dx = (x) + 2g(x)) of R6 Average(f ) · 1If 0 (f (x) Average(f · g) 6 and 0 (2f the points x = 2 ,= 2 , 3 , 0 (f (x) g(x)) 4 . (c) Estimate 0 (x +1) dx using a right-hand sum with Z 3 (b) Use your graph to decide which of yourfour numbers 1 19 f (x) n = 3. Represent this sum on the sketch. Is this (b) (2 pts) Figure 5.38 4x3 + g(x)dx (x x) Z n sum an overestimate or underestimate? 3 sin(x2 ) dx n = 1, 2, 3, 4 1 5-2h31 5-4h44figa 5-4h45 5-4h46 x x 5. (2 pts) Explain what is wrong with the following calculation of the area under the curve x4 from x = 0 5-4h44figb 1 2 1 2 Z 1 h is largest. Which is smallest? How many of the num1 1 i1 1 1 2 Figure 5.72 Figure 5.73 bers 3 positive? are dx = = = . x4 3x 3 3 3 1 1 45. (a) Without computing any integrals, explain why the ins5-4h54-56 average value of f (x) the area [0, ] the curve 6. (3 pts) Suppose = sin x on undermust be between= 0 and x = b. Solve for a in terms of b. x 0.5 to 1. (b) Compute this average. 46. Figure ?? shows the standard normal distribution from statistics, which is given by 2 1 e x /2 . 2 Statistics books often contain tables such as the following, which show the area under the curve from 0 to b for various values of b. R For Problems ??, assuming F = f , mark the quantity on a 1 to x = 1: x ecopy of Figure0??. x = a is six times the area under the curve 2e2x from from x = to (in other words, write a =(some formula involving b)) F (x) ins5-4h54-56fig x a b Figure 5.75 Z 5 1 dx = ln(5). Now find a fraction which approximates ln(5), by x 1 Z 5 1 using M4 (midpoint sum with 4 rectangles) to approximate dx. 1 x (The actual value of ln(5) is 1.6094 . . .. For fun, plug your approximation into a calculator and compare) 7. (4 pts) Use the Evaluation Theorem to show 6000 and the supply curve is given by P = Q + 10. Find Q + 50 the equilibrium price and quantity, and compute the consumer and producer surplus. 8. (5 pts) Suppose the demand curve is given by P = Some extra practice (not to be handed in) Z 3 1. Estimate f (x) dx using R5 and L5 . −2 −1 1 2 3 −5 −4 −3 −2 −3 −1 1 2 3 4 5 2 2. Suppose h is a function such that h(1) = 2, h0 (1) = 3, h00 (1) = 4, h(2) = 6, h0 (2) = 5, h00 (2) = 13, and h00 is R2 continuous everywhere. Find 1 h00 (u) du. ...
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