HW5-solutions - 290 5-2h21 Chapter Five KEY CONCEPT THE...

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Unformatted text preview: 290 5-2h21 Chapter Five KEY CONCEPT: THE DEFINITE INTEGRAL 21. (a) Find the total area between f (x) = x3 − x and5-2h32 the 32. Use Figure ?? to find the values of x-axis! for 0 ≤ x ≤ 3. "2 "7 3 Mathematics for Economists (a) fII (x) dx (b) f (x) dx 0 3 f (x)dx. (b) Find "7 "8 Homework 5 (c) f (x) dx (d) f (x) dx 0 2 5 (c) Are the answers parts date (a) and of (b) the same? ExDue ontothe next lecture: April 20th. 2016. plain. Queries: [email protected] 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 22. 23. 24. 25. 26. 27. 28. 29. 5-2h34 Area = 13 ! a b 5-2h29fig "Area = 2 30. Given 0 cos √ x dx and interpret " 2π 5-2h36 36. Estimate ! (a) "1 0 2 e−x dx using n = 5 rectangles to form a Left-hand sum (b) Right-hand sum 5-2h37 "0 f (x)dx = 4 and Figure ??, estimate: −2 "2 f (x)dx (b) (a) 0 Solution: (c) The total shaded area. "2 f (x)dx −2 The area between the fgraph of f (x) x = b is 13, so 2 Z b x −2 2 a −2 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 and the x-axis between x = a and terms in the sum, but do not evaluate it. (c) Which sum is an overestimate? Which sum is an underestimate? f (x) dx = 13. 5-2h38 38. (a) Draw the rectangles " π that give the left-hand sum approximation to 0 sin x dx with n = 2. Since the graph of f (x) is below the x-axis for b < x < c, " 0 (b) Repeat part (a) for −π sin x dx. Z c (c) From your answers to parts (a) and (b), what is Figure 5.37 " π value of the left-hand sum approximation to f (x) dx = −2. the "0 5-2h30fig 5-2h31fig "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 Under y = 6x3 − 2 for 5 ≤ x ≤ 10. −1 Under the curve y = cos t for 0 ≤ t ≤ π/2. 1. Use the Figure below to find the values of 5-2h32fig −2 Under y = ln x for 1 ≤ x ≤ 4. 2 4 6 8 10 R b Under y = 2 cos(t/10) for 1 ≤ t ≤ 2. Figure 5.39: Graph consists of a semicircle and (a) a f (x) dx √ Under the curve line segments R c y = cos x for 0 ≤ x ≤ 2. (b)curveb yf (x) dxx2 and above the x-axis. Under the =7− R 4 c = x − 8 and below the x-axis. Above the curve (c) a fy (x) dx Use Figure ?? find the values of R to 5-2h33 33. (a) Graph f (x) = x(x + 2)(x − 1). c " (d) b |f (x)| dx(b) " c f (x) dx a f (x) dx (a) (b) Find the total area between the graph and the x-axis " ac "bc between (c) f (x) dx (d) |f (x)| dx " 1 x = −2 and x = 1. a a (c) Find −2 f (x) dx and interpret it in terms of areas. f (x) 5-2h30 2 sin x dx with n = 4? 31. (a) Using Figure ??, find −3 f (x) dx. −π b " (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 Since the graph of f (x) is above the x-axis for a < x < b and below for f (x) dx. −3 Represent"this value as the area under a curve. 6 b < x < c,1 Z (b) Estimate 0 (x2 + 1) dx using a left-hand sum with f (x)c = 3. Represent this sum graphically on a sketch f4(x) dx = 13 − 2 =n11. x of f (x) = x2 + 1. Is this sum an overestimate or a −4 −3 −2 −1 1 2 3 5 underestimate the true value found in part (a)? " 6 of −1 2 The graph of |f (x)| is the same as the graph of f (x) except thata right-hand the (c) Estimate (x +1) dx using sum with 0 = 3.Thus Represent this sum on your sketch. Is this 5.38 part belowFigure the x-axis is reflected to be aboven it. sum an overestimate or underestimate? Z b |f (x)| dx = 13 + 2 = 15. a 1 2. Using the graph of f in Figure, arrange the following quantities in increasing order, from least to greatest. Provide your reasoning. Z 1 f (x) dx (i) 0 Z 2 f (x) dx (ii) 1 Z 2 f (x) dx (iii) 0 Z 3 f (x) dx (iv) 2 Z 2 f (x) dx (v) − 1 (vi) the number 0 5-4h43 0 !2 0 !2 2 f (x)(viii) dx the number " 4 20 5-4h47 47. (vi) The number 0 (viii) The number −10 f (x) 10 1 5-4h43fig !13 (iv) f (x) dx (v) − 1 f (x) dx (vii) The number 20 2 ins5-4h49-52 3 −10 x ! Figure 5.71 5-4h44figa 5-4h46 5-4h49 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 44. (a) Using Figures ?? and ??, find the average value on " 5 " 7 0 ≤Solution: x ≤ 2 of f (x + 2) dx (f (x) + 2) dx R1 R 2 5-4h51 51. R 3 5-4h52 52. (i) Since f (x) (ii) g(x)= A (iii) f (x)·g(x) −2 2 and 0 f (x) dx and f (x) dx = −A f (x) dx = A 1 3 , we 0 1 2 (b) Is the following statement true? Explain your anknow that swer. 5-4h53 Z 1 Z 2 53. (a) SketchZ a3graph√of f√ (x) =√sin(x√2 ) and mark on it Average(f ) · Average(g) = Average(f · g) the points x = π, 2π, 3π, 4π. 1 5-4h45 311 43. Using the graph of f in Figure ??, arrange the ins5-4h47-48 following In Problems ??, evaluate the expression, if possible, or (vii) the number − 10 quantities in increasing order, from least to greatest. say ! 4 what additional information is needed, given that !1 !2 g(x) dx = 12. f (x) dx (ii) f (x) dx (i) −4 (iii) 5-4h44 5.4 THEOREMS ABOUT DEFINITE INTEGRALS 45. (a) f (x) dx < − 0< 1 f (x) 0 g(x) f (x) dx < 1 R2 f (x) dx. (b) Use your graph to decide which of the four numbers 2 " √ nπ In addition,x 0 f (x) dx = A1 −x A2 , which is negative,sin(x but2 ) dx smaller n = 1,in 2, 3, 4 R2 0 5-4h44figb 1 2 1 2 magnitude than 1 f (x) dx. Thus is largest. Which is smallest? How many of the numFigure 5.72 Figure Z 2 5.73 Z 2 bers are positive? f (x)ins5-4h54-56 dx < f (x) dx < 0. # Without computing any integrals, explain why the For Problems ??, assuming F = f , mark the quantity on a average value of f (x) = sin x on 1[0, π] must be be- 0copy of Figure ??. tween 0.5 and 1. The area A3 lies inside a rectangle of height 20 and base 1, so A3 < 20. (b) Compute this average. The area A2 liesnormal insidedistribution a rectangle below the x-axis of height F (x) 10 and 46. Figure ?? shows the standard from width 1, so −10 < A . Thus: statistics, which is given by 2 1 −x2 /2 . (viii) √<2π e(ii) < (iii) < (vi) < (i) < (v) < ( iv) < (vii). Statistics books often contain tables such as the following, which show the area under the curve from 0 to b for x ins5-4h54-56fig 2 a b various values of b. ! Area = √1 !b Figure 5.75 e−x 2 /2 dx 3. Evaluate each integral by representing it in terms of areas. NOTE: Do not be tempted to write down an answer using the formulas you know to do this (you can use them to check your final answer though) R2 (a) −1 (1 − x) dx √ R0 (b) −3 (1 + 9 − x2 ) dx R2 (c) −1 |x| dx 1Solution: ! 1 ! 4. Find a function f and a number a such that Z x √ f (t) dt = 2 x 6+ 2 t a for all x > 0. [ Hint: Use the Fundamental Theorem of Calculus (FTC)] Solution: ! 3 5. Compute the following integrals: (a) (b) R4 √ e√ x 1 x dx R2 √ x x − 1 dx 1 Solution: (a) Let √ x = u. Then 1 √ dx 2 x Z = du, hence √ e x √ dx = 2 x Z √1 dx x = 2du. We get eu du = 2eu = 2e √ x . So the integral is 2e2 − 2e. (b) Let x − 1 = u. Then dx = du. We have R√ R u(u + 1) du = u3/2 + u1/2 du = 25 u5/2 + 23 u3/2 . Back substituting we get: 2 (x − 1)5/2 + 32 (x − 1)3/2 . Evaluating at the limits we have: 5 2 + 32 . 5 4 ...
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