MFE2-hw7 - Math for Econ II Written Assignment 7(28 points...

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Unformatted text preview: Math for Econ II, Written Assignment 7 (28 points) Due Friday, April 4th New York University 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. • Staple your homework if you have multiple pages! 1. Assume an economics model where we have three industries 0 0.1 C = 0.5 0.4 0.1 0.2 and final demand with the following consumption matrix 0.4 0.2 , 0.3 4 d = 8 , 6 where the components of d are in millions of dollars. (a) (2 pts) What does the entry a21 = 0.5 mean? (b) (2 pts) Compute the output levels to meet the total demand. Note: You can use a calculator to do the computations for problem 1. 290 5-2h21 ins5-2h22-28 5-2h22 5-2h23 5-2h24 5-2h25 5-2h26 5-2h27 5-2h28 5-2h29 Chapter Five KEY CONCEPT: THE DEFINITE INTEGRAL 21. (a) Find the total area between f (x) = x3 − x and5-2h32 32. Use Figure ?? to find the values of the x-axis for 0 ≤ x ≤ 3. 2 7 3 2. (8 points total; (a) and (b) are 2 points, while (c) is worth 4dx (a) f (x) points) (b) f (x) dx 0 3 f (x)dx. (b) Find 12 7 8 1 (c) f (x) dx (d) f (x) dx 0 2 5 dx using Estimate (c) Are the answers to parts (a) and (b) the same? Exx+1 3 plain. (a) a left-hand sum with n = 3. Is this estimate an over or under estimate? 2 In Problems ??, find the area of the regions between the curve f (x) (b) a right-hand sum with n = 3. Is this estimate an over or under estimate? and the horizontal axis 1 22. 23. 24. 25. 26. 27. 28. 29. x (c) Write for formula for Under y = 6x3 − 2the 5 ≤ x ≤ 10. Rn (the right-endpoint sum with n rectangles) using sum notation. −1 Under the curve y = cos t for 0 ≤ t ≤ π/2. 5-2h32fig −2 Under y = ln x for 1 ≤ x ≤ 4. 2 4 6 8 10 Under y = 2 cos(t/10) for 1 ≤ t ≤ 2. Figure 5.39: Graph consists of a semicircle and √ 3. (1 pt each) Use x figure x ≤ 2. Under the curve y = cos the for 0 ≤ below to find the values of line segments Under the curve y = 7 − x2 and above the x-axis. b (a) a f (x) dx Above the curve y = x4 − 8 and below the x-axis. c (b) b f find the Use Figure ?? to (x) dx values of 5-2h33 33. (a) Graph f (x) = x(x + 2)(x − 1). b c c f af (b) f (x) dx (a) (c)(x) dx(x) dx (b) Find the total area between the graph and the x-axis a b c c between x = −2 and x = 1. (c) f (x)c |f (x)| dx (d) dx |f (x)| dx a a 1 (d) a (c) Find −2 f (x) dx and interpret it in terms of areas. f (x) 5-2h34 Area = 13 © 34. Compute the definite integral the result in terms of areas. 4 0 cos √ x dx and interpret 5-2h35 a b 5-2h29fig 35. Without computation, decide if 0 e−x sin x dx is positive or negative. [Hint: Sketch e−x sin x.] 5-2h36 36. Estimate x c sArea = 2 ! (a) Figure 5.36 5-2h30 30. Given 5-2h37 0 f (x)dx = 4 and Figure ??, estimate: −2 2 f (x)dx (b) (a) 0 (c) The total shaded area. 2 −2 f (x)dx f (x) 2π 1 0 2 e−x dx using n = 5 rectangles to form a Left-hand sum (b) 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 the right Riemann sum 2 with n = 2 approximating 1 ln x dx. Write out the terms in the sum, but do not evaluate it. 4. (4 pts) Find a function f so that the limit below is equal to n 2+ lim n→∞ i=1 2√ 5. (4 pts) Find the exact values of 0 4 − x2 dx and Now use properties of the definite integral to find 5 2 f (x) dx. 3i 2+ 3i 3 ·e n · . n n 2 (2 0 − x) dx by using areas of familiar geometric shapes. 2 ( 4 − x2 + 2 − x) dx 0 6. (4 pts) Find the area under the curve y = x2 between x = 0 and x = 1 by finding a general formula for Ln (the left-hand sum using n rectangles) and computing limn→∞ Ln . When doing this problem, you will find the formula below useful: n−1 i2 = i=1 n(n − 1)(2n − 1) 6 ...
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