# hwk-8-mfe-2 - plain ins5-2h22-28 In Problems nd the area of...

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Unformatted text preview: plain. ins5-2h22-28 In Problems ??, ﬁnd the area of the regions between the curve and the horizontal axis 5-2h22 5-2h23 5-2h24 5-2h25 5-2h26 5-2h27 5-2h28 5-2h29 2 f (x) 1 Math for Econ II, Written Assignment 8 (28 points) 3 x 22. Under y = 6x − 2 for 5 ≤ x ≤ 10. Due Friday, April 17 −1 23. Please the curve y = solutions ≤ t ≤ π/2. Under write neat cos t for 0 for the problems below. Show all your work. If you only write the answer with no work, 5-2h32fig −2 24. you will = ln be given x ≤ 4. Under y not x for 1 ≤ any credit. 2 4 6 8 10 25. 26. 27. 28. 29. Under y = 2 cos(t/10) for 1 ≤ t ≤ 2. √ • Write your name and recitation section number. Figure 5.39: Graph consists of a semicircle and Under the curve y = cos x for 0 ≤ x ≤ 2. line segments Under the curve y =homework above the x-axis. • Staple your 7 − x2 and if you have multiple pages! Above the curve y = x4 − 8 and below the x-axis. 1. (2 pts total; 1 pt each) Use the ﬁgure below to ﬁnd the values of Use Figure ?? to ﬁnd the values of 5-2h33 33. (a) Graph f (x) = x(x + 2)(x − 1). b c f bf (b) f (x) dx (a) (b) Find the total area between the graph and the x-axis (a)(x) dx(x) dx a b c c c between x = −2 and x = 1. (c) f (x)c |f (x)| dx (d) dx |f (x)| dx a a 1 (b) a (c) Find −2 f (x) dx and interpret it in terms of areas. f (x) √ b 5-2h29fig c 34. Compute the deﬁnite integral the result in terms of areas. 5-2h35 35. Without computation, decide if 0 e−x sin x dx is positive or negative. [Hint: Sketch e−x sin x.] 5-2h36 © a 4 5-2h34 Area = 13 36. Estimate x sArea = 2 ! cos x dx and interpret 2π (a) Figure 5.36 0 1 0 2 e−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 0 graph of f in FigureFigure ??, estimate: following f (x)dx = 4 and ??, arrange the ins5-4h47-48 In Problems ??, evaluate = ln x, represent the left Riemann 5-4h43 43. Using the 5-2h30 30. Given 2 or sum with quantitiesexpression, if ln x dx. Write least to greatest. n = 2the in increasingpossible,from approximating −2 2. (4 pts) Using the graph of f in Figure, arrange the following order, quantities2in increasing order, from least to greatest. say what out the terms in the sum, but do not1evaluate it. additional information is needed, given that 2 2 2 3 2 4 (x)dx f (a) 1 0 f 1 2 i) (x) f (x) dx ii) (b) (x)(x)(x)dx iv) = 12. (b) On another sketch, − 1 f (x) dx vi) The number 0 f total (ii) f (i)(c) 0 The 0 dx shaded area. 1 f1 −2dxdx iii) 0 f (x) dx g(x) dx 2 f (x) dx v) represent the right Riemann sum −4 2 2 3 with n = 2 approximating 1 ln x dx. Write out the (iii) f (x) dx (iv) f (x) dx 4 4 0 2 f (x) terms in the sum, but do not evaluate it. 2 5-4h48 48. 20 viii) The 0 5-4h47 47. g(x) dx g(−x) dx (v) − vii) (x) dx f The number (vi) The numbernumber −10 (c) Which sum is an overestimate? Which sum is an un1 2 0 −4 (vii) The number 20 (viii) The number −10 derestimate? 10 −2 5-2h30fig 5-4h43fig −10 −2 1 x f (x) 2 2 5-2h38 ins5-4h49-52 3 Figure 5.37 x ! 38. (a) Draw the rectangles that give the left-hand sum apπ proximation to 0 expression if possible, In Problems ??, evaluate the sin x dx with n = 2. or say 7 0 what extra information is needed, given dx. (x) dx = 25. (b) Repeat part (a) for −π sin x 0 f (c) From your answers to parts (a) and (b), what is sum approximation to 7 the value of the left-hand 3.5 π 5-4h50 50. 49. f−π sin x dx with n = 4? (x) dx f (x) dx 0 Figure 5.71 5-4h49 31. (a) Using Figure ??, ﬁnd −3 f (x) dx. 0 0 (b) If the area of the shaded region is A, estimate 5-4h44 44. (a) Using Figures ?? and ??, ﬁnd the average value on 5-2h39 39. (a) Use a calculator or computer to ﬁnd 6 (x2 + 1) dx. 4 −1 4 4 5 7 dx. 3. ≤−3 f (x)of dx = f (x) dx = value as the ﬁnd 1 f0(x) dx, 5. 0 (3 pts) Suppose f is even, −2 f (x) 5-4h51 3, and 2Represent this5-4h52Either area under a curve. or show there is not x≤2 51. f (x + 2) dx 6 2 52. (f (x) + 2) dx 4 (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 1 −2 0 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 1 2 3 55-4h53 53. (a) Sketch a graph of f (x) = sin(x2 ) and mark on it underestimate of the true value√ found in part (a)? √ √ 5-2h31fig Average(f )−1If 1 (f (x) − 2g(x)) dxg) 6 and 1 (2f the points x = √π,= 2π,ﬁnd 1 (f (x) − g(x)) dx. · 6 dx (x) + 2g(x))(x2 +1)9, using 0 right-hand sum with (a) (3 pts) Average(g) = Average(f · = 0 (c) Estimate 0 dx 3π, a 4π. 0 3 (b) Use your graph to decide which of yourfour numbers 1 n = 3. Represent this sum on the sketch. Is this Figure19 5.38 f (x) g(x) 3 (b) (2 pts) (x − 4x + x) dx √ sum an overestimate or underestimate? nπ −3 sin(x2 ) dx n = 1, 2, 3, 4 5-2h31 5-4h44figa x Figure 5.72 5-4h45 5-4h46 x 0 5-4h44figb 5. (2 pts) Explain what is wrong with2the following calculation of the area under the curve 1 2 1 Figure 5.73 1 Area = √1 b −x2 /2 e from x = −1 to x = 1: is largest. Which is smallest? How many of the num- 1 1 1 1 1 2 bers are dx = − 3 positive? − = − . =− 4 x 3x −1 3 3 3 −1 45. (a) Without computing any integrals, explain why the ins5-4h54-56 average value of f (x) = sin x on [0, π] must be be6. (3 0.5 and 1. tween pts) Suppose the area under the curve x = 0 to x = b. (b) Compute this average. Solve for a in terms of b. 46. Figure ?? shows the standard normal distribution from statistics, which is given by 2 1 √ e−x /2 . 2π 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. ' 1 x4 For Problems ??, assuming F = f , mark the quantity on a 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 a b Figure 5.75 dx x 5 1 dx = ln(5). Now ﬁnd a fraction which approximates ln(5), by x 1 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) 3 f (x) dx using R5 and L5 . 1. Estimate −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, h (1) = 3, h (1) = 4, h(2) = 6, h (2) = 5, h (2) = 13, and h is 2 continuous everywhere. Find 1 h (u) du. ...
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