Hw7 - Math for Econ II Written Assignment 7(28 points Due...

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Calculus: Early Transcendentals
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Chapter 5 / Exercise 54
Calculus: Early Transcendentals
Stewart
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Unformatted text preview: Math for Econ II, Written Assignment 7 (28 points) Due Friday, April 7 Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you will not be given any credit. • Write your name and recitation section number. 5.4 THEOREMS ABOUT DEFINITE INTEGRALS • Staple your homework if you have multiple pages! 5-4h43 43. Using the graph of f in Figure ??, arrange the ins5-4h47-48 following In Problems ??, evaluate the expression, if possible, or 1. (4inpts) Usingorder, the graph of fto below, the following quantities in increasing order, from least to greatest. quantities increasing from least greatest.arrange say information ! 4 what Radditional R 2 is needed, given that R2! R2 3 !1 R1 2 g(x) i)f (x)0 dx f (x) dx ii)(ii) 1 f (x) dxdx iii) 0 f (x) dx iv)dx 2=f12. (x) dx v) − 1 f (x) dx vi) The number 0 f (x) (i) −4 0 (iii) !2 0 (iv) f (x) dx !13 f (x) dx 2 !2 5-4h47 Thedxnumber (vi) 20 The viii)number The number −10 47. (v) − vii) f (x) 0 1 (vii) The number 20 (viii) 1 5-4h43fig The number −10 f (x) 10 ins5-4h49-52 2 3 −10 x ! Figure 5.71 5-4h44 5-4h49 (x − x + x − x) dx −1 5-4h44figa 4 5-4h48 g(x) dx 48. 0 " 4 g(−x) dx −4 In Problems ??, evaluate the expression ! 7if possible, or say what extra information is needed, given 0 f (x) dx = 25. 49. #" 7 5-4h50 f (x) dx 50. 0 " 3.5 f (x) dx 0 " √ nπ sin(x2 ) dx n = 1, 2, 3, 4 x 4. (2 pts) A5-4h44figb honeybee population startsx with 100 bees and increases at a rate of n0 (t) bees per week. What does 0 1 R 15 2 1 2 0 100 + 0 n (t) dt represent? is largest. Which is smallest? How many of the num- Figure 5.72 5-4h46 " R −1 value on R4 44. (a) Using Figures ?? and ??, find the average " 5 R 4 f (x) dx = 5. Either" find 7 2. dx = 3, and f (x) dx, or show there is not 0 ≤(3x pts) ≤ 2 ofSuppose f is even, R−2 f (x) 5-4h51 2 1 5-4h52 51. f (x + 2) dx 52. (f (x) + 2) dx 4 enough information to find f (x) dx. (i) f (x) (ii) g(x) (iii)1 f (x)·g(x) −2 0 (b) Is the following statement true? Explain your an3. (5 pts) Evaluate the following integrals: swer. 2 5-4h53 53. R1 R 1(a) Sketch a graph of f√ R√ =√sin(x 1 ) and mark on it √ =(x) (a) (3 pts) If 0 (f (x)=−Average(f 2g(x)) dx = 6 and 0 (2f the (x)points + 2g(x)) 9, find (x) − g(x)) dx. Average(f ) · Average(g) · g) 3π, 0 (f π, 2π, 4π. x = dx Z 1 (b) Use your graph to decide which of the four numbers 1 117 f (x) g(x) 5 3 (b) (2 pts) 5-4h45 311 Figure 5.73 bers are positive? 5. (2 pts) Explain what is wrong with the following calculation of the area under the curve x12 from x = −1 to x = 1: 45. (a) Without computing any integrals, explainins5-4h54-56 whyZ the 1 h 1??, i 1assuming F # = f , mark the quantity on a For Problems average value of f (x) = sin x on [0, π] must be be- 1 copy dx = − ??. of Figure = −1 − 1 = −2. 2 x −1 tween 0.5 and 1. −1 x (b) Compute this average. x 6. ?? (3 shows pts) Suppose thenormal area under the curve twenty times the area under the curve 2e2x (x) 46. Figure the standard distribution from e from x = 0 to x = a is F from x is =given 0 to by x = b. Solve for a in terms of b. (in other words, write a =(some formula involving b)) statistics, which 2 1 √ e−x /2 . 2π Z 5 Statistics books often contain tables such as the follow1 7. (4 pts) Use the Evaluation Theorem to show dx = ln(5). Now find a fraction which approximates ln(5), by ing, which show the area under the curve from 0 to b for x x 1ins5-4h54-56fig Z a5 b various values of b. 1 using M4 (midpoint sum with 4 rectangles) to approximate dx. Figure 5.75 x 1 ! b −x2 /2 1 ! of (The actual value . . .. For fun, plug your approximation into a calculator and compare) Arealn(5) = √ is 1.6094 e dx 2π 0 5-4h46fig x 5-4h54 54. A slope representing f (a). " b 6000 8. (5 pts) Suppose the demand curve is5-4h55 given55. by AP length = representing and the fsupply (x) dx. curve is given by P = Q + 10. Find Q + 50 Figure 5.74 the equilibrium price and quantity, and compute the consumer anda producer surplus. " b ! 1 b −x2 /2 5-4h56 1 b 0 b √ 2π 0 e 1 0.3413 2 0.4772 3 0.4987 4 0.5000 56. A slope representing dx 5-4h57 Use the information given in the table and the symmetry of the curve about the y-axis to find: " 3 " 3 2 2 1 1 e−x /2 dx (b) √ e−x /2 dx (a) √ b−a f (x) dx. a 57. In Chapter ??, the average velocity over the time interval a ≤ t ≤ b was defined to be (s(b) − s(a))/(b − a), where s(t) is the position function. Use the Fundamental Theorem of Calculus to show that the average value of the velocity function v(t), on the interval a ≤ t ≤ b, is also (s(b) − s(a))/(b − a). ins5-2h22-28 5-2h22 5-2h23 5-2h24 5-2h25 5-2h26 5-2h27 5-2h28 5-2h29 In Problems ??, find the area of the regions between the curve and the horizontal axis 5-2h34 Area = 13 ! a b 5-2h29fig "Area = 2 30. Given "0 "2 "2 −2 ! (a) 5-2h30fig −2 Figure 5.37 5-2h31fig "0 x dx and interpret " 2π "1 0 2 e−x dx using n = 5 rectangles to form a (b) Left-hand sum Right-hand sum 37. (a) On a sketch of y = ln x, represent"the left Riemann 2 sum with n = 2 approximating 1 ln x dx. Write out the terms in the sum, but do not evaluate it. (b) On another sketch, represent " 2the right Riemann sum with n = 2 approximating 1 ln x dx. Write out the terms in the sum, but do not evaluate it. (c) Which sum is an overestimate? Which sum is an underestimate? 5-2h38 38. (a) Draw the rectangles " π that give the left-hand sum approximation to 0 sin x dx with n = 2. f (x)dx x √ 5-2h37 2 2 cos 36. Estimate f (x) −2 0 5-2h36 f (x)dx = 4 and Figure ??, estimate: −2 f (x)dx (b) (a) 0 (c) The total shaded area. "4 35. Without computation, decide if 0 e−x sin x dx is positive or negative. [Hint: Sketch e−x sin x.] x c 34. Compute the definite integral the result in terms of areas. 5-2h35 Figure 5.36 5-2h31 f (x) 1 x 22. Under y = 6x3 − 2 for 5 ≤ x ≤ 10. Some extra practice (not to be handed in) −1 23. Under the curve y = cos t for 0 ≤ t ≤ π/2. 5-2h32fig0 −2 24. Under y = ln x for ≤ 4. 1. Suppose h is1 ≤ ax function such that h(1) = −2, h (1) = 3, h002(1) = 44, h(2)6 = 6, 8h0 (2) 10 = 5, h00 (2) = 13, and h00 is R 2 00 25. Undercontinuous y = 2 cos(t/10) for 1 ≤ t ≤ 2. everywhere. Find 1 h (u) du. Figure 5.39: Graph consists of a semicircle and √ 26. Under the curve y = cos x for 0 ≤ x ≤ 2. line segments 27. Under the curve y = 7 − x2 and above the x-axis. 28. Above the curve y = x4 − 8 and below the x-axis. 2. Use the figure below to find the values of 29. Use Figure ?? to find the values of 5-2h33 33. (a) Graph f (x) = x(x + 2)(x − 1). "b Rb "c f (x) dx (b) f (x) dx (a) (a) f (x) dx (b) Find the total area between the graph and the x-axis " ac Rc "bc between c (c) f (x) dx " 1 x = −2 and x = 1. a (b) |f (x)| dx (d) a |f (x)| dx a (c) Find −2 f (x) dx and interpret it in terms of areas. f (x) 5-2h30 2 "0 (b) Repeat part (a) for −π sin x dx. (c) From your answers to parts (a) and (b), what is "theπ value of the left-hand sum approximation to sin x dx with n = 4? −π 31. (a) Using Figure ??, find −3 f (x) dx. " (b) " If the area of the shaded region is A, estimate 5-2h39 39. (a) Use a calculator or computer to find 6 (x2 + 1) dx. 4 0 f (x) dx. −3 Represent"this value as the area under a curve. 6 (b) Estimate 0 (x2 + 1) dx using a left-hand sum with 1 f (x) n = 3. Represent this sum graphically on a sketch 4 x of f (x) = x2 + 1. Is this sum an overestimate or −4 −3 −2 −1 1 2 3 5 underestimate " 6 of the true value found in part (a)? −1 (c) Estimate 0 (x2 +1) dx using a right-hand sum with n = 3. Represent this sum on your sketch. Is this Figure 5.38 sum an overestimate or underestimate? ...
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