textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Dashboard X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks BusCalc.pdf 178 / 272 69% + 174 14. What are the units for the "area" of a rectangle with the given base and height units? Base units Height units "Area" units miles per second seconds hours dollars per hour square feet feet kilowatts hours houses people per house meals meals In problems 15-17, represent the area of each bounded region as a definite integral, and use geometry to determine the value of the definite integral. 15. The region bounded by y = 2x , the x-axis, the line x = 1, and x = 3. 175 16. The region bounded by y = 4-2x, the x-axis, and the y-axis. 17. The shaded region in the graph to the right. y = 3 - 2 176 Chapter 3 The Integral Business Calculus 178 area = 5 18. Using the graph of f shown and the given areas of area = 2 several regions, evaluate (a) Jf(x) dx ( b) f f ( x) dx ( c ) f f ( x ) dx 0 3 5 area = 3 19. Using the graph of f shown and the given areas of several regions, evaluate: (a) J g(x) dx b) J g(x) dx area = 6 3 177 8 (c) J g(x) dx (d) J g(x) dx area 20. Use the graph to evaluate: (a) h(x) dx (b) S h(x) dx -2 4 c J h(x) dx (d) J h(x) dx -2 -2 178 21. Your velocity along a straight road is shown to the right.textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Dashboard X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks BusCalc.pdf 179 / 272 69% 175 Chapter 3 The Integral Business Calculus 179 b In problems 23 - 26, the units are given for x and for f(x) . Give the units of J f(x) dx . 176 23. x is time in "seconds", and f(x) is velocity in "meters per second." 24. x is time in "hours", and f(x) is a flow rate in "gallons per hour." 25. x is a position in "feet", and f(x) is an area in "square feet." 26. x is a position in "inches", and f(x) is a density in "pounds per inch." In problems 27 -31, represent the area with a definite integral and use technology to find the approximate answer. 27. The region bounded by y = x , the x-axis, the line x = 1, and x = 5. 177 28. The region bounded by y = Vx , the x-axis, and the line x = 9. 29. The shaded region shown to the right. y = Vx 30. The shaded region below. 31. Consider the definite integral | (3 + x) dx . 178 (a) Using six rectangles, find the left-hand Riemann sum for this definite integral. b) Using six rectangles, find the right-hand Riemann sum for this definite integral. c) Using geometry, find the exact value of this definite integral. 32. Consider the definite integral x' dx. (a) Using four rectangles, find the left-hand Riemann sum for this definite integral. (b) Using four rectangles, find the right-hand Riemann sum for this definite integral. 33. Write the total distance traveled by the car in the graph between 1 pm and 4 pm as a definite integral and estimate the value of the integral. velocity (miles/hour) 20 179textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Dashboard X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks BusCalc.pdf 180 / 272 69% + 176 c) Using geometry, find the exact value of this definite integral. 32. Consider the definite integral | x' dx. (a) Using four rectangles, find the left-hand Riemann sum for this definite integral. (b) Using four rectangles, find the right-hand Riemann sum for this definite integral 33. Write the total distance traveled by the car in the graph between 1 pm and 4 pm as a definite integral and estimate the value of the integral. velocity (miles/hour) 177 Chapter 3 The Integral Business Calculus 180 areas Problems 34-41 refer to the graph of f shown. Use the y= f(x) graph to determine the values of the definite integrals. The bold numbers represent the area of each region.) 2 178 34. J f(x) dx 35. J f(x) dx 36. J f(x) dx 0 3 2 7 5 37. J f(w) dw 38. J f(x) dx 39. J f(x) dx 40. [ f(t) at 41 . J f ( x ) dx 6 3 Problems 42-47 refer to the graph of g shown. Use the graph to evaluate the integrals y = g(x) 42. J g(x) dx 43. J g(t) dt 44. J g(x) dx 179 45. J g(s) ds 46. f 2g(1) at 47. S 1+g(x) dx 5 AVA 180textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Dashboard X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks E BusCalc.pdf 187 / 272 69% + Chapter 3 The Integral Business Calculus 187 3.2 Exercises In problems 1 - 5, verify that F(x) is an antiderivative of the integrand f(x) and use Part 2 of the Fundamental Theorem to evaluate the definite integrals. 1 . f 2 x dx , F ( x ) = 1 2 + 5 2 . f 3 x dx , F ( x ) = x 3 + 2 3 . fx ax , F( 1 ) = 3 x3 187 4 . S ( 12 + 4x - 3 ) dx , F ( x ) - 3 x3 + 212 - 3x 5 . fax , F( x ) = In( x ) 0 6. Given A(x) = S 2t dt, find A'(x) 0 7. Given A(x) = S (3-12) dt, find A'(x) y = f(t) 188 8. Let A(x) = J f(f) at for the function graphed here. Evaluate A'(1), A'(2) , A' ( 3 ) . KA For problems 9-10, the graph provided shows g'(x). Use it sketch a graph of g(x) that satisfies g(0) = 0. 189 9. 10. 190textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Dashboard X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks E BusCalc.pdf 194 / 272 69% + Chapter 3 The Integral Business Calculus 194 3.3 Exercises For problems 1-10, find the indicated antiderivative 1. f(x3 -14x + 5)dx 2. [ (2.5x5 - x-1.25) dx 3. 12.3dy 4. [n'dw 193 5. fedp 6. [ Vitex - 1 dx 7. d 8. J dx 9. [ (x - 2)x + 2)dx 10. [ - at For problems 11-18, find an antiderivative of the integrand and use the Fundamental Theorem to evaluate the definite integral 5 11. J 3x2 dx 12. J x2 dx 13. S ( x2 + 4x - 3) dx 14. x dx 2 194 15 . [ xdx 16 . [ Vx dx 17 . 1 4 dx 18 . 1 1 dx - For problems 19 - 21 find the area shown in the figure. y= 4 3 + y = 1 +* y = (x- 2)2 y =x2 19 . 20. 21. 195 196textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Dashboard X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks E BusCalc.pdf 200 / 272 69% + 197 Chapter 3 The Integral Business Calculus 200 3.4 Exercises For problems 1-8, find the indicated antiderivative. ( 4x + 1) 2. Je 10ox dx 3. (1.0003) 12 dt 4. felix 5. [Vw+5 dw 5. fox' V3x3 - 1 dx 7. fax 198 xInx Jx 2 - 6x +5 For problems 9-12, find an antiderivative of the integrand and use the Fundamental Theorem to evaluate the definite integral 9. ( 2x dx 10. fezx dx 11. S (x-2)3 dx 12. J x VI-x2 dx 2 199 200textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (ONL) X BusCalc.pdf X *Dashboard X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks BusCalc.pdf 211 / 272 69% + Chapter 3 The Integral Business Calculus 211 3.6 Exercises In problems 1 - 4, use the values in the table to estimate the areas. f (x g(x) h(x) 209 3 6 2 N W A a a un 5 6 1. Estimate the area between fand g, between x = 0 and x = 4. 2. Estimate the area between g and h, between x = 0 and x = 6. 3. Estimate the area between fand h, between x = 0 and x = 4. 4. Estimate the area between fand g, between x = 0 and x = 6. Island 5. Estimate the area of the island shown - 340 feet_ _- 410 feet - 220 feet 240 feet_ In problems 6 - 15, find the area between the graphs of f 210 and g for x in the given interval. Remember to draw the graph! - 100 6 6. f(x) =x2 + 3, g(x) = 1 and -1 Sx$2. 7. f( x) = x2+3, g(x) = 1 + x and OSx53. 8 . f( x ) = x2 , g(x ) = x and 0Sx52. 9 . f ( x ) = ( x -1)2 , g(x) = x + 1 and OSx53. 10. f(x) = > , g(x) =x and 1 SxSe. 1 1. f(x) = Vx , g(x) =x and 0 Sx 54. 217 12 . f ( x ) = 4 -x 2, g(x) = x + 2 and OSx52. 13 . f (x ) = et , g(x) = x and OSx52 O 14 . f(x ) = 3 , g(x) = V/1 -x2 and OSx51. 15 . f(x ) = 2, g(x) = V/4-x2 and -25x $2. I'M 212textbroker.com X Final Exam X X Canvas LMS X MATH140-12141 (0 X BusCalc.pdf X *Dashboard X Meeting 19 - Chapt X - CD A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks BusCalc.pdf 218 / 272 69% + Chapter 3 The Integral Business Calculus 218 From t = 0 to t = 1 (the first year), the income is constant $7000 per year. From t = 1 to t = 8, the income is increasing by $800 each year; the income flow function F(t) will be F(t)=7000 + 800(t -1)= 6200 + 800t. To find the present value, we'll have to divide the integral into the two pieces, one for each of the functions: 216 PV = [7000e 017'dt + (6200 + 800t)e 0.017'dt = 70166. The present value is greater than the cost of the machine, so the company should buy the machine. 3.7 Exercises 1. The demand and supply functions for a certain product are given by p = 150-.5q and p =.002q2 +1.5, where p is in dollars and q is the number of items. (a) Which is the demand function? (b) Find the equilibrium price and quantity 217 (c) Find the total gains from trade at the equilibrium price. 2. Still thinking about the product from Exercise 1, with its demand and supply functions, suppose the price is set artificially at $70 (which is above the equilibrium price) a) Find the quantity supplied and the quantity demanded at this price. (b) Compute the consumer surplus at this price, using the quantity demanded (c) Compute the producer surplus at this price, using the quantity demanded (why?). (d) Find the total gains from trade at this price. (e) What do you observe? 3. When the price of a certain product is $40, 25 items can be sold. When the price of the same product costs $20, 185 items can be sold. On the other hand, when the price of this product is $40, 200 items will be produced. But when the price of this product is $20, only 100 items will be produced. Use this information to find supply and demand functions (assume for simplicity that the functions are linear), and compute the consumer and producer surplus at 218 the equilibrium price. 4. Find the present and future values of a continuous income stream of $5000 per year for 12 years if money can earn 1.3% annual interest compounded continuously. 5. Find the present value of a continuous income stream of $40,000 per year for 35 years if money can earn (a) 0.8% annual interest, compounded continuously, (b) 2.5% annual interest, compounded continuously (c) 4.5% annual interest, compounded continuously. 219textbroker.com X -Final Exam X X Canvas LMS X MATH140-12141 (0 X BusCalc.pdf X *Dashboard X Meeting 19 - Chapt X -> C A Not Secure | opentextbookstore.com/buscalc/BusCalc.pdf Update : Suggested Sites M Gmail YouTube Disney+ canvas find cards saved o... Other Bookmarks E BusCalc.pdf 228 / 272 69% + 3.8 Exercises 225 In problems 1 -4, check that the function y is a solution of the given differential equation. 1. y' + 3y = 6. y=e-3X +2. 2. y' - 2y =8. y=e2X -4. 3. y' =-x/y. y= V7-x2 . 4. y' = x-y. y=x-1+2e-* . 226 Chapter 3 The Integral Business Calculus 228 In problems 5 -8 check that the function y is a solution of the given initial value problem. 5. y' = 6x2 - 3 and y(1) =2. y= 2x5 - 3x + 3. 5. y' = 6x + 4 and y(2) =3. y=3x2+ 4x - 17. 7. y' = by and y(0) =7. y= 75x 8. y' =-2y and y(0) =3. y= 3e-2x 227 In problems 9- 12, a family of solutions of a differential equation is given. Find the value of the constant C so the solution satisfies the initial value condition. 9. y' = 2x and y(3) =7. y =x2+ C. 10. y' = 3x2- 5 and y(1) = 2. y = x3 - 5x + C. 11. y' = 3y and y(0) = 5. y = Ce3x 11. y' =-2y and y(0) = 3. y = Ce-2x In problems 13 - 18, solve the differential equation. (Assume that x and y are restricted so that division by zero does not occur.) 13. y' = 2x 14. y' = x/y 15. xy' = y + 3 228 16 . y' = x y + 3y 17. y' = 4y 18. y' = 5(2 -y) In problems 19-22, solve the initial value separable differential equations. 19. y' = 2xy for y(0) = 3, y(0) = 5, and y(1) = 2. 20. y' = x/y for y(0) = 3, y(0) = 5, and y(1) = 2. 21. y' = 3y for y(0) = 4, y(0) = 7, and y(1) = 3. 22. y' =-2y for y(0) = 4, y(0) = 7, and y(1) = 3. 23. The rate of growth of a population P(t) which starts with 3,000 people and increases by 4% per year is P '(t) = 0.0392-P(t). Solve the differential equation and use the solution to estimate the population