Explanation Given Information: The curve is f ( x ) = x 2 + x + 1 from x = − 1 to x = 1 and 4 subintervals and the function is evaluated at left end points of the subintervals. Formula used: The base of the rectangles to approximate the area is b − a n where the interval [ a , b ] on which function is defined is divided into n subintervals. The height of the rectangles is the value of the function calculated at the left-hand end point of the interval containing the base. The area of a rectangle is base × height . The approximated area under the curve is the sum of the areas of each rectangle. The value of the sum ∑ k = 1 n k 3 = ( n ( n + 1 ) 2 ) 2 . Calculation: The curve is f ( x ) = x 2 + x + 1 from x = − 1 to x = 1 and 4 subintervals and the function is evaluated at left end points of the subintervals. The base of the rectangles to approximate the area is b − a n where the interval [ a , b ] on which function is defined is divided into n subintervals. Since, the function is defined from x = − 1 to x = 1 . So, a = − 1 and b = 1 . Thus, Base of each rectangle = 1 − ( − 1 ) 4 = 2 4 = 0.5 Thus, the 4 subintervals, each of length 0.5 are [ − 1 , − 0.5 ] , [ − 0.5 , 0 ] , [ 0 , 0.5 ] and [ 0.5 , 1 ] . Since, there are 4 subintervals, the number of rectangles is 4. Recall that the height of the rectangles is the value of the function calculated at the left end point of the interval containing the base. Record these values in a table. Rectangle Base Left endpoint Height Area = b a s e × h e i g h t 1 0

Mathematical Applications for the Management, Life, and Social Sciences

11th Edition

ISBN: 9781305108042

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Chapter 13.1, Problem 8E

To determine

To calculate: The area under the graph of f(x)=x2+x+1 from x=−1 to x=1 using the 4 equal-intervals of the same size and use rectangles to find the exact area.

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Chapter 13.1, Problem 7E

Chapter 13.1, Problem 9E

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Chapter 13 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 13.1 - Prob. 1CP Ch. 13.1 - Prob. 2CP Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E

Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E Ch. 13.1 - Prob. 11E Ch. 13.1 - Prob. 12E Ch. 13.1 - Prob. 13E Ch. 13.1 - Prob. 14E Ch. 13.1 - In Problems 14-19, find the value of each sum. 15....Ch. 13.1 - Prob. 16E Ch. 13.1 - Prob. 17E Ch. 13.1 - Prob. 18E Ch. 13.1 - Prob. 19E Ch. 13.1 - Prob. 20E Ch. 13.1 - Prob. 21E Ch. 13.1 - Prob. 22E Ch. 13.1 - Prob. 23E Ch. 13.1 - Prob. 24E Ch. 13.1 - Prob. 25E Ch. 13.1 - Prob. 26E Ch. 13.1 - Prob. 27E Ch. 13.1 - Prob. 28E Ch. 13.1 - Prob. 29E Ch. 13.1 - Prob. 30E Ch. 13.1 - Prob. 31E Ch. 13.1 - Prob. 32E Ch. 13.1 - Prob. 33E Ch. 13.1 - Prob. 34E Ch. 13.1 - Prob. 35E Ch. 13.1 - 36. Speed trials The figure gives the time it...Ch. 13.1 - Prob. 37E Ch. 13.1 - 38. Drug levels in the blood The manufacturer of a...Ch. 13.1 - Prob. 39E Ch. 13.1 - Prob. 40E Ch. 13.2 - CHECKPOINT 1. True or false: (a) For any integral,...Ch. 13.2 - Prob. 2CP Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 2E Ch. 13.2 - Prob. 3E Ch. 13.2 - Prob. 4E Ch. 13.2 - Prob. 5E Ch. 13.2 - Prob. 6E Ch. 13.2 - Prob. 9E Ch. 13.2 - Prob. 10E Ch. 13.2 - Prob. 11E Ch. 13.2 - Prob. 12E Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 8E Ch. 13.2 - Prob. 23E Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 14E Ch. 13.2 - Prob. 15E Ch. 13.2 - Prob. 16E Ch. 13.2 - Prob. 17E Ch. 13.2 - Prob. 18E Ch. 13.2 - Prob. 19E Ch. 13.2 - Prob. 20E Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 22E Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 26E Ch. 13.2 - Prob. 27E Ch. 13.2 - Prob. 28E Ch. 13.2 - Prob. 29E Ch. 13.2 - Evaluate the definite integrals in Problems...Ch. 13.2 - Prob. 31E Ch. 13.2 - Prob. 32E Ch. 13.2 - Prob. 33E Ch. 13.2 - Prob. 34E Ch. 13.2 - Prob. 35E Ch. 13.2 - Prob. 36E Ch. 13.2 - In Problems 39-42, (a) write the integral that...Ch. 13.2 - Prob. 38E Ch. 13.2 - In Problems 39-42, (a) write the integral that...Ch. 13.2 - In Problems 39-42, (a) write the integral that...Ch. 13.2 - Prob. 41E Ch. 13.2 - 44. Find the area between the curve and the...Ch. 13.2 - Prob. 43E Ch. 13.2 - 46. Find the area between the curve and the...Ch. 13.2 - Prob. 45E Ch. 13.2 - Prob. 46E Ch. 13.2 - Prob. 47E Ch. 13.2 - Prob. 48E Ch. 13.2 - Prob. 49E Ch. 13.2 - Prob. 50E Ch. 13.2 - Prob. 51E Ch. 13.2 - Prob. 52E Ch. 13.2 - 55. Depreciation The rate of depreciation of a...Ch. 13.2 - Depreciation The rate of depreciation of a...Ch. 13.2 - 57. Sales and advertising A store finds that its...Ch. 13.2 - Prob. 56E Ch. 13.2 - 59. Total income The income from an oil change...Ch. 13.2 - 60. Total income Suppose that a vending machine...Ch. 13.2 - Prob. 59E Ch. 13.2 - Prob. 60E Ch. 13.2 - Prob. 61E Ch. 13.2 - Prob. 62E Ch. 13.2 - Prob. 63E Ch. 13.2 - Prob. 64E Ch. 13.2 - 67. Testing The time t (in minutes) needed to read...Ch. 13.2 - 68. Response time In a small city, the response...Ch. 13.2 - Customer service The duration t (in minutes) of...Ch. 13.2 - 70. Product life The useful life of a car battery...Ch. 13.3 - CHECKPOINT 1. True or false: (a) Over the interval...Ch. 13.3 - Prob. 2CP Ch. 13.3 - Prob. 3CP Ch. 13.3 - For each shaded region in Problems 1-6, (a) form...Ch. 13.3 - For each shaded region in Problems 1-6, (a) form...Ch. 13.3 - For each shaded region in Problems 1-6, (a) form...Ch. 13.3 - For each shaded region in Problems 1-6, (a) form...Ch. 13.3 - Prob. 5E Ch. 13.3 - For each shaded region in Problems 1-6, (a) form...Ch. 13.3 - For each shaded region in Problems 7-12, (a) find...Ch. 13.3 - For each shaded region in Problems 7-12, (a) find...Ch. 13.3 - Prob. 9E Ch. 13.3 - Prob. 10E Ch. 13.3 - For each shaded region in Problems 7-12, (a) find...Ch. 13.3 - Prob. 12E Ch. 13.3 - In Problems 13-26, equations are given whose...Ch. 13.3 - Prob. 14E Ch. 13.3 - In Problems 13-26, equations are given whose...Ch. 13.3 - Prob. 16E Ch. 13.3 - Prob. 17E Ch. 13.3 - Prob. 18E Ch. 13.3 - Prob. 19E Ch. 13.3 - Prob. 20E Ch. 13.3 - Prob. 21E Ch. 13.3 - Prob. 22E Ch. 13.3 - Prob. 23E Ch. 13.3 - In Problems 13-26, equations are given whose...Ch. 13.3 - Prob. 25E Ch. 13.3 - Prob. 26E Ch. 13.3 - Prob. 27E Ch. 13.3 - Prob. 28E Ch. 13.3 - Prob. 29E Ch. 13.3 - In Problems 27-32, find the average value of each...Ch. 13.3 - Prob. 31E Ch. 13.3 - Prob. 32E Ch. 13.3 - Prob. 33E Ch. 13.3 - Prob. 34E Ch. 13.3 - Prob. 35E Ch. 13.3 - 36. Sales and advertising The figure shows the...Ch. 13.3 - Prob. 37E Ch. 13.3 - 38. Inventory management The figure shows how an...Ch. 13.3 - 39. Sales and advertising The number of daily...Ch. 13.3 - 40. Demand The demand function for a certain...Ch. 13.3 - 41. Social Security beneficiaries With data from...Ch. 13.3 - Prob. 42E Ch. 13.3 - Prob. 43E Ch. 13.3 - Prob. 44E Ch. 13.3 - Prob. 45E Ch. 13.3 - 46. Income distribution In an effort to make the...Ch. 13.3 - Prob. 47E Ch. 13.3 - Prob. 48E Ch. 13.3 - Prob. 49E Ch. 13.3 - Prob. 50E Ch. 13.4 - CHECKPOINT 1. Suppose that a continuous income...Ch. 13.4 - Prob. 2CP Ch. 13.4 - CHECKPOINT 3. If demand is , supply is , and the...Ch. 13.4 - 1. Find the total income over the next 10 years...Ch. 13.4 - 2. Find the total income over the next 8 years...Ch. 13.4 - 3. Suppose that a steel company views the...Ch. 13.4 - 4. Suppose that the Quick-Fix Car Service...Ch. 13.4 - 5. A small brewery considers the output of its...Ch. 13.4 - 6. A company that services a number of vending...Ch. 13.4 - 7. A franchise models the profit from its store as...Ch. 13.4 - 8. The Medi Spa franchise has a continuous income...Ch. 13.4 - 9. A continuous income stream has an annual rate...Ch. 13.4 - 10. A continuous income stream has an annual rate...Ch. 13.4 - 11. The income from an established chain of...Ch. 13.4 - Prob. 12E Ch. 13.4 - 13. Suppose that a printing firm considers its...Ch. 13.4 - 14. Suppose that a vending machine company is...Ch. 13.4 - 15. A 58-year-old couple are considering opening a...Ch. 13.4 - Prob. 16E Ch. 13.4 - Prob. 17E Ch. 13.4 - Prob. 18E Ch. 13.4 - In Problems 17-26, p and C are in dollars and x is...Ch. 13.4 - Prob. 20E Ch. 13.4 - Prob. 21E Ch. 13.4 - Prob. 22E Ch. 13.4 - In Problems 17-26, p and C are in dollars and x is...Ch. 13.4 - Prob. 24E Ch. 13.4 - In Problems 17-26, p and C are in dollars and x is...Ch. 13.4 - Prob. 26E Ch. 13.4 - 27. Suppose that the supply function for a good...Ch. 13.4 - Prob. 28E Ch. 13.4 - 29. If the supply function for a commodity is ,...Ch. 13.4 - Prob. 30E Ch. 13.4 - Find the producers surplus at market equilibrium...Ch. 13.4 - 32. Find the producer's surplus at market...Ch. 13.4 - Prob. 33E Ch. 13.4 - Prob. 34E Ch. 13.4 - 35. The demand function for a certain product is ...Ch. 13.4 - Prob. 36E Ch. 13.5 - Prob. 1CP Ch. 13.5 - Prob. 2CP Ch. 13.5 - Prob. 3CP Ch. 13.5 - Prob. 4CP Ch. 13.5 - Prob. 5CP Ch. 13.5 - Prob. 6CP Ch. 13.5 - Prob. 1E Ch. 13.5 - Prob. 2E Ch. 13.5 - Prob. 3E Ch. 13.5 - Prob. 4E Ch. 13.5 - Prob. 5E Ch. 13.5 - Prob. 6E Ch. 13.5 - Prob. 7E Ch. 13.5 - Prob. 8E Ch. 13.5 - Prob. 9E Ch. 13.5 - Prob. 10E Ch. 13.5 - Prob. 11E Ch. 13.5 - Prob. 12E Ch. 13.5 - Prob. 13E Ch. 13.5 - Prob. 14E Ch. 13.5 - Prob. 15E Ch. 13.5 - Prob. 16E Ch. 13.5 - Prob. 17E Ch. 13.5 - Prob. 18E Ch. 13.5 - Evaluate the integrals in Problems 1-32. Identify...Ch. 13.5 - Prob. 20E Ch. 13.5 - Prob. 21E Ch. 13.5 - Prob. 22E Ch. 13.5 - Prob. 23E Ch. 13.5 - Prob. 24E Ch. 13.5 - Prob. 25E Ch. 13.5 - Prob. 26E Ch. 13.5 - Prob. 27E Ch. 13.5 - Prob. 28E Ch. 13.5 - Prob. 29E Ch. 13.5 - Prob. 30E Ch. 13.5 - Prob. 31E Ch. 13.5 - Prob. 32E Ch. 13.5 - Prob. 33E Ch. 13.5 - Prob. 34E Ch. 13.5 - Prob. 35E Ch. 13.5 - Prob. 36E Ch. 13.5 - Prob. 37E Ch. 13.5 - Prob. 38E Ch. 13.5 - Prob. 39E Ch. 13.5 - Prob. 40E Ch. 13.5 - 41. Income streams Suppose that when a new oil...Ch. 13.5 - Prob. 42E Ch. 13.6 - Prob. 1CP Ch. 13.6 - Prob. 2CP Ch. 13.6 - Prob. 1E Ch. 13.6 - Prob. 2E Ch. 13.6 - Prob. 3E Ch. 13.6 - Prob. 4E Ch. 13.6 - Prob. 5E Ch. 13.6 - Prob. 6E Ch. 13.6 - Prob. 7E Ch. 13.6 - Prob. 8E Ch. 13.6 - Prob. 9E Ch. 13.6 - Prob. 10E Ch. 13.6 - Prob. 11E Ch. 13.6 - In Problems 1-16, use integration by parts to...Ch. 13.6 - Prob. 13E Ch. 13.6 - Prob. 14E Ch. 13.6 - Prob. 15E Ch. 13.6 - Prob. 16E Ch. 13.6 - Prob. 17E Ch. 13.6 - Prob. 18E Ch. 13.6 - Prob. 19E Ch. 13.6 - Prob. 20E Ch. 13.6 - Prob. 21E Ch. 13.6 - Prob. 22E Ch. 13.6 - Prob. 23E Ch. 13.6 - Prob. 24E Ch. 13.6 - Prob. 25E Ch. 13.6 - Prob. 26E Ch. 13.6 - Prob. 27E Ch. 13.6 - Prob. 28E Ch. 13.6 - Prob. 29E Ch. 13.6 - Prob. 30E Ch. 13.6 - Prob. 31E Ch. 13.6 - Prob. 32E Ch. 13.6 - Prob. 33E Ch. 13.6 - Prob. 34E Ch. 13.6 - Prob. 35E Ch. 13.6 - Prob. 36E Ch. 13.6 - Prob. 37E Ch. 13.6 - Prob. 38E Ch. 13.7 - Prob. 1CP Ch. 13.7 - Prob. 2CP Ch. 13.7 - In Problems 1-20, evaluate the improper integrals...Ch. 13.7 - Prob. 2E Ch. 13.7 - In Problems 1-20, evaluate the improper integrals...Ch. 13.7 - Prob. 4E Ch. 13.7 - Prob. 5E Ch. 13.7 - Prob. 6E Ch. 13.7 - Prob. 7E Ch. 13.7 - Prob. 8E Ch. 13.7 - Prob. 9E Ch. 13.7 - Prob. 10E Ch. 13.7 - Prob. 11E Ch. 13.7 - Prob. 12E Ch. 13.7 - Prob. 13E Ch. 13.7 - Prob. 14E Ch. 13.7 - Prob. 15E Ch. 13.7 - Prob. 16E Ch. 13.7 - Prob. 17E Ch. 13.7 - Prob. 18E Ch. 13.7 - Prob. 19E Ch. 13.7 - Prob. 20E Ch. 13.7 - Prob. 21E Ch. 13.7 - 22. For what value of c does Ch. 13.7 - Prob. 23E Ch. 13.7 - Prob. 24E Ch. 13.7 - Prob. 25E Ch. 13.7 - Prob. 26E Ch. 13.7 - Prob. 27E Ch. 13.7 - Prob. 28E Ch. 13.7 - Prob. 29E Ch. 13.7 - Prob. 30E Ch. 13.7 - 31. Find the value of c so that is a probability...Ch. 13.7 - Prob. 32E Ch. 13.7 - Prob. 33E Ch. 13.7 - Prob. 34E Ch. 13.7 - 35. Find the area below the graph of and above...Ch. 13.7 - Prob. 36E Ch. 13.7 - Prob. 37E Ch. 13.7 - 38. Capital value Suppose that a donor wants to...Ch. 13.7 - 39. Capital value Suppose that a business provides...Ch. 13.7 - Prob. 40E Ch. 13.7 - Prob. 41E Ch. 13.7 - 42. Capital value Suppose that a business provides...Ch. 13.7 - 43. Repair time In a manufacturing process...Ch. 13.7 - 44. Customer service The duration t (in minutes)...Ch. 13.7 - Prob. 45E Ch. 13.7 - Prob. 47E Ch. 13.8 - CHECKPOINT 1. Suppose [1, 4] is divided into 6...Ch. 13.8 - Prob. 2CP Ch. 13.8 - Prob. 3CP Ch. 13.8 - Prob. 1E Ch. 13.8 - Prob. 2E Ch. 13.8 - Prob. 3E Ch. 13.8 - Prob. 4E Ch. 13.8 - Prob. 5E Ch. 13.8 - Prob. 6E Ch. 13.8 - Prob. 7E Ch. 13.8 - Prob. 8E Ch. 13.8 - Prob. 9E Ch. 13.8 - Prob. 10E Ch. 13.8 - Prob. 11E Ch. 13.8 - Prob. 12E Ch. 13.8 - Prob. 13E Ch. 13.8 - Prob. 14E Ch. 13.8 - Prob. 15E Ch. 13.8 - Prob. 16E Ch. 13.8 - Prob. 17E Ch. 13.8 - Prob. 18E Ch. 13.8 - Prob. 19E Ch. 13.8 - Prob. 20E Ch. 13.8 - Prob. 21E Ch. 13.8 - Prob. 22E Ch. 13.8 - In Problems 23-30, round all calculations to 2...Ch. 13.8 - In Problems 23-30, round all calculations to 2...Ch. 13.8 - Prob. 25E Ch. 13.8 - Prob. 26E Ch. 13.8 - In Problems 23-30, round all calculations to 2...Ch. 13.8 - In Problems 23-30, round all calculations to 2...Ch. 13.8 - Prob. 29E Ch. 13.8 - Prob. 30E Ch. 13.8 - Prob. 31E Ch. 13.8 - Prob. 32E Ch. 13.8 - Pollution monitoring Suppose that the presence of...Ch. 13.8 - Prob. 34E Ch. 13 - 1. Calculate . Ch. 13 - 2. Use formulas to simplify Ch. 13 - Prob. 3RE Ch. 13 - Prob. 4RE Ch. 13 - 5. Use a definite integral to find the area under...Ch. 13 - Prob. 6RE Ch. 13 - Prob. 7RE Ch. 13 - Prob. 8RE Ch. 13 - Prob. 9RE Ch. 13 - Prob. 10RE Ch. 13 - Prob. 11RE Ch. 13 - Prob. 12RE Ch. 13 - Prob. 13RE Ch. 13 - Prob. 14RE Ch. 13 - Prob. 15RE Ch. 13 - Prob. 16RE Ch. 13 - Prob. 17RE Ch. 13 - Prob. 18RE Ch. 13 - Prob. 19RE Ch. 13 - Prob. 20RE Ch. 13 - Prob. 21RE Ch. 13 - Prob. 22RE Ch. 13 - Prob. 23RE Ch. 13 - Prob. 24RE Ch. 13 - Prob. 25RE Ch. 13 - Prob. 26RE Ch. 13 - Prob. 27RE Ch. 13 - Prob. 28RE Ch. 13 - Prob. 29RE Ch. 13 - In Problems 27-30, use integration by parts to...Ch. 13 - Prob. 31RE Ch. 13 - Prob. 32RE Ch. 13 - Prob. 33RE Ch. 13 - Prob. 34RE Ch. 13 - Prob. 35RE Ch. 13 - Prob. 36RE Ch. 13 - 38. Use the table that follows to...Ch. 13 - 39. Suppose that a definite integral is to be...Ch. 13 - 40. Maintenance Maintenance costs for buildings...Ch. 13 - Prob. 40RE Ch. 13 - Prob. 41RE Ch. 13 - Prob. 42RE Ch. 13 - Prob. 43RE Ch. 13 - Prob. 44RE Ch. 13 - Prob. 45RE Ch. 13 - Prob. 46RE Ch. 13 - 46. Income streams Suppose that a machine’s...Ch. 13 - Prob. 48RE Ch. 13 - Prob. 49RE Ch. 13 - Prob. 50RE Ch. 13 - Prob. 51RE Ch. 13 - 54. Capital value Find the capital value of a...Ch. 13 - 55. Total income Suppose that a continuous income...Ch. 13 - Prob. 55RE Ch. 13 - 1. Use left-hand endpoints and subdivisions to...Ch. 13 - 2. Consider from with n equal subdivisions. (a)...Ch. 13 - Prob. 3T Ch. 13 - Prob. 4T Ch. 13 - Prob. 5T Ch. 13 - Prob. 6T Ch. 13 - Prob. 7T Ch. 13 - Prob. 8T Ch. 13 - 9. Suppose the supply function for a product is ...Ch. 13 - 10. Suppose a continuous income stream has an...Ch. 13 - Prob. 11T Ch. 13 - Prob. 12T Ch. 13 - Prob. 13T Ch. 13 - Prob. 14T Ch. 13 - Prob. 15T Ch. 13 - 16. The environmental effects of a chemical spill...

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The area under the graph of f ( x ) = x 2 + x + 1 from x = − 1 to x = 1 using the 4 equal-intervals of the same size and use rectangles to find the exact area.

Solution Summary: The author calculates the area under the graph of f(x) = 2+x+1 using the 4 equal-intervals of the same size and uses rectangles to find the exact

To calculate: The area under the graph of f(x)=x2+x+1 from x=−1 to x=1 using the 4 equal-intervals of the same size and use rectangles to find the exact area.

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Chapter 13 Solutions