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12th Edition

Ronald J. Harshbarger + 1 other

Publisher: Cengage Learning

ISBN: 9781337625340

Chapter 11.3, Problem 48E

Textbook Problem

At what points does the curve defined by

(a) horizontal tangents?

(b) vertical tangents?

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 11.1 - 1.
Ch. 11.1 - 2. If
Ch. 11.1 - 3. If
Ch. 11.1 - 4. Find .
Ch. 11.1 - Find the derivatives of the functions in Problems...Ch. 11.1 - Find the derivatives of the functions in Problems...Ch. 11.1 - Find the derivatives of the functions in Problems...Ch. 11.1 - Find the derivatives of the functions in Problems...Ch. 11.1 - Find the derivatives of the functions in Problems...Ch. 11.1 - Find the derivatives of the functions in Problems...

Ch. 11.1 - Find the derivatives of the functions in Problems...Ch. 11.1 - Find the derivatives of the functions in Problems...Ch. 11.1 - Find the derivatives of the functions in Problems...Ch. 11.1 - Find the derivatives of the functions in Problems...Ch. 11.1 - 11. Find .
Ch. 11.1 - 12. Find .
Ch. 11.1 - In each of Problems 13-18, find the derivative of...Ch. 11.1 - In each of Problems 13-18, find the derivative of...Ch. 11.1 - In each of Problems 13-18, find the derivative of...Ch. 11.1 - In each of Problems 13-18, find the derivative of...Ch. 11.1 - In each of Problems 13-18, find the derivative of...Ch. 11.1 - In each of Problems 13-18, find the derivative of...Ch. 11.1 - 19. Find .
Ch. 11.1 - 20. Find .
Ch. 11.1 - 21. Find
Ch. 11.1 - 22. Find
Ch. 11.1 - 23. Find
Ch. 11.1 - 24. Find
Ch. 11.1 - In Problems 25-38, find y'.
Ch. 11.1 - In Problems 25-38, find y'.
26.
Ch. 11.1 - In Problems 25-38, find y'.
27.
Ch. 11.1 - In Problems 25-38, find y'.
28.
Ch. 11.1 - In Problems 25-38, find y'.
29.
Ch. 11.1 - In Problems 25-38, find y'.
30.
Ch. 11.1 - In Problems 25-38, find y'.
31.
Ch. 11.1 - In Problems 25-38, find y'.
32.
Ch. 11.1 - In Problems 25-38, find y'.
33.
Ch. 11.1 - In Problems 25-38, find y'.
34.
Ch. 11.1 - In Problems 25-38, find y'.
35.
Ch. 11.1 - In Problems 25-38, find y'.
36.
Ch. 11.1 - In Problems 25-38, find y'.
37.
Ch. 11.1 - In Problems 25-38, find y'.
38.
Ch. 11.1 - In Problems 39-42, find the relative maxima and...Ch. 11.1 - In Problems 39-42, find the relative maxima and...Ch. 11.1 - In Problems 39-42, find the relative maxima and...Ch. 11.1 - In Problems 39-42, find the relative maxima and...Ch. 11.1 - 43. Marginal cost Suppose that the total cost (in...Ch. 11.1 - 44. Investment If money is invested at the...Ch. 11.1 - 45. Marginal revenue The total revenue, in...Ch. 11.1 - 46. Supply Suppose that the supply of x units of a...Ch. 11.1 - 47. Demand The demand function for a product is...Ch. 11.1 - 48. pH level The pH of a solution is given...Ch. 11.1 - 49. Drug concentration Concentration (in mg/ml) in...Ch. 11.1 - 50. Decibels The loudness of sound (L, measured in...Ch. 11.1 - 51. Richter scale The Richter scale reading, R,...Ch. 11.1 - 52. Women in the workforce From 1950 and projected...Ch. 11.2 - 1. If , find y’.
Ch. 11.2 - 2. If , find y’.
Ch. 11.2 - 3. If find ds/dt.
Ch. 11.2 - 4. If the sales of a product are given by , where...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - Find the derivatives of the functions in Problems...Ch. 11.2 - 35. (a) What is the slope of the line tangent to...Ch. 11.2 - 36. (a) What is the slope of the line tangent to
...Ch. 11.2 - 37. The equation for the standard normal...Ch. 11.2 - 38. (a) Find the mode of the normal distribution*...Ch. 11.2 - In Problems 39-42, find any relative maxima and...Ch. 11.2 - In Problems 39-42, find any relative maxima and...Ch. 11.2 - In Problems 39-42, find any relative maxima and...Ch. 11.2 - In Problems 39-42, find any relative maxima and...Ch. 11.2 - 43. Future value If $P is invested for n years at...Ch. 11.2 - 44. Future value The future value that accrues...Ch. 11.2 - 45. Sales decay After the end of an advertising...Ch. 11.2 - 46. Sales decay The sales decay for a product is...Ch. 11.2 - 47. Marginal cost Suppose that the total cost in...Ch. 11.2 - 48. Marginal revenue Suppose that the revenue in...Ch. 11.2 - 49. Drugs in a bloodstream The percent...Ch. 11.2 - 50. Radioactive decay The amount of the...Ch. 11.2 - 51. Pollution Pollution levels in Lake Sagamore...Ch. 11.2 - 52. Drug concentration Suppose the concentration ,...Ch. 11.2 - 53. National health care With U.S. Department of...Ch. 11.2 - 54. Personal consumption Using US. Bureau of Labor...Ch. 11.2 - 55. Richter scale The intensity of an earthquake...Ch. 11.2 - 56. Decibel readings The intensity level of sound,...Ch. 11.2 - 58. Blood pressure Medical research has shown...Ch. 11.2 - 59. Spread of disease Suppose that the spread of a...Ch. 11.2 - 60. Spread of a rumor The number of people in a...Ch. 11.2 - Population Using Social Security Administration...Ch. 11.2 - 62. Carbon dioxide emissions Using U.S. Department...Ch. 11.2 - 65. Disposable income per capita Using U.S. Energy...Ch. 11.2 - 66. Energy use U.S. energy use per dollar of GDP...Ch. 11.3 - Find the following:
(b) (c)
Ch. 11.3 - 2. Find .
Ch. 11.3 - In Problems 1-6, find dy/dx at the given point...Ch. 11.3 - In Problems 1-6, find at the given point without...Ch. 11.3 - In Problems 1-6, find dy/dx at the given point...Ch. 11.3 - In Problems 1-6, find at the given point without...Ch. 11.3 - In Problems 1-6, find at the given point without...Ch. 11.3 - In Problems 1-6, find at the given point without...Ch. 11.3 - Find dy/dx for the functions in Problems 7-10.
7....Ch. 11.3 - Find for the functions in Problems 7-10.
8.
Ch. 11.3 - Find for the functions in Problems 7-10.
9.
Ch. 11.3 - Find for the functions in Problems 7-10.
10.
Ch. 11.3 - 11.
Ch. 11.3 - 12.
Ch. 11.3 - 13.
Ch. 11.3 - 14.
Ch. 11.3 - 15.
Ch. 11.3 - 16.
Ch. 11.3 - 17.
Ch. 11.3 - 18. If find .
Ch. 11.3 - 19.
Ch. 11.3 - 20.
Ch. 11.3 - 21. Find for
Ch. 11.3 - 22. Find y' for
Ch. 11.3 - For Problems 23-26, find the slope of the line...Ch. 11.3 - For Problems 23-26, find the slope of the line...Ch. 11.3 - For Problems 23-26, find the slope of the line...Ch. 11.3 - For Problems 23-26, find the slope of the line...Ch. 11.3 - In Problems 27-30, write the equation of the...Ch. 11.3 - In Problems 27-30, write the equation of the...Ch. 11.3 - In Problems 27-30, write the equation of the...Ch. 11.3 - In Problems 27-30, write the equation of the...Ch. 11.3 - 31. If ln find .
Ch. 11.3 - 32. If ln find .
Ch. 11.3 - 33. If , find .
Ch. 11.3 - 34. If ln find .
Ch. 11.3 - 35.
Ch. 11.3 - 36.
Ch. 11.3 - 37.
Ch. 11.3 - 38.
Ch. 11.3 - 39.
Ch. 11.3 - 40.
Ch. 11.3 - 41. Find the slope of the tangent to the curve
Ch. 11.3 - 42. Write the equation of the line tangent to the...Ch. 11.3 - 43. Find the slope of the line tangent to the...Ch. 11.3 - 44. Find the slope of the line tangent to the...Ch. 11.3 - 45. Write the equation of the line tangent to the...Ch. 11.3 - 46. Write the equation of the line tangent to the...Ch. 11.3 - 47. At what points does the curve defined by ...Ch. 11.3 - 48. At what points does the curve defined by ...Ch. 11.3 - 49. In Problem 11, the derivative y' was found to...Ch. 11.3 - 50. (a) Find y' implicitly for (b) By taking...Ch. 11.3 - 51. Find and simplify.
Ch. 11.3 - 52. Find y" for
Ch. 11.3 - In Problems 53 and 54, find the maximum and...Ch. 11.3 - In Problems 53 and 54, find the maximum and mini-...Ch. 11.3 - 55. Advertising and sales Suppose that a company’s...Ch. 11.3 - 56. Insect control Suppose that the number of...Ch. 11.3 - 57. Production Suppose that a company can produce...Ch. 11.3 - 58. Production Suppose that production of 10,000...Ch. 11.3 - 59. Demand If the demand function for q units of a...Ch. 11.3 - 60. Demand If the demand function for units of a...Ch. 11.3 - 61. Radioactive decay The number of grams of...Ch. 11.3 - 62. Disease control Suppose the proportion P of...Ch. 11.3 - 63. Summer Simmer Index In the late 1980s, J. Pepi...Ch. 11.4 - 1. If V represents volume, write a mathematical...Ch. 11.4 - 2. (a) Differentiate implicitly with respect to...Ch. 11.4 - 3. True or false: In solving a related-rates...Ch. 11.4 - In Problems 1-4, find using the given values.
Ch. 11.4 - In Problems 1-4, find using the given values.
2....Ch. 11.4 - In Problems 1-4, find using the given values.
3....Ch. 11.4 - In Problems 1-4, find using the given values.
4....Ch. 11.4 - In Problems 5-8, assume that x and y are...Ch. 11.4 - In Problems 5-8, assume that x and y are...Ch. 11.4 - In Problems 5-8, assume that x and y are...Ch. 11.4 - In Problems 5-8, assume that x and y are...Ch. 11.4 - 9.
Ch. 11.4 - 10.
.
Ch. 11.4 - 11. A point is moving along the graph of the...Ch. 11.4 - 12. A point is moving along the graph of the...Ch. 11.4 - 13. The radius of a circle is increasing at a rate...Ch. 11.4 - 14. The area of a circle is changing at a rate of...Ch. 11.4 - 15. The volume of a cube is increasing at a rate...Ch. 11.4 - 16. The lengths of the edges of a cube are...Ch. 11.4 - 17. Profit Suppose that the daily profit (in...Ch. 11.4 - 18. Profit Suppose that the monthly revenue and...Ch. 11.4 - 19. Demand Suppose that the price p (in dollars)...Ch. 11.4 - 20. Supply The supply function for a product is...Ch. 11.4 - 21. Capital investment and production Suppose that...Ch. 11.4 - 22. Boyle's law Boyles law for enclosed gases...Ch. 11.4 - Tumor growth For Problems 23 and 24, suppose that...Ch. 11.4 - Tumor growth For Problems 23 and 24, suppose that...Ch. 11.4 - 25. Allomelric relationships—fish For many species...Ch. 11.4 - 26. Blood flow The resistance R of a blood vessel...Ch. 11.4 - 27. Allometric relationships—crabs For fiddler...Ch. 11.4 - 28. Body weight and surface area For human beings,...Ch. 11.4 - 29. Cell growth A bacterial cell has a spherical...Ch. 11.4 - 30. Water purification Assume that water is being...Ch. 11.4 - 31. Volume and radius Suppose that air is being...Ch. 11.4 - 32. Boat docking Suppose that a boat is being...Ch. 11.4 - 33. Ladder safety A 30-ft ladder is leaning...Ch. 11.4 - 34. Flight A kite is 30 ft high and is moving...Ch. 11.4 - 35. Flight A plane is flying at a constant...Ch. 11.4 - 36. Distance Two boats leave the same port at the...Ch. 11.4 - 37. Distance Two cars are approaching an...Ch. 11.4 - 38. Water depth Water is flowing into a barrel in...Ch. 11.4 - 39. Water depth Suppose that water is being pumped...Ch. 11.5 - 1. Write the formula for point elasticity, .
Ch. 11.5 - 2. (a) If , the demand is called _______.
(b) If...Ch. 11.5 - 3. Find the elasticity of demand tor when p = 10...Ch. 11.5 - 4. For problems involving taxation in a...Ch. 11.5 - In Problems 1 -8, p is in dollars and q is the...Ch. 11.5 - In Problems 1 -8, p is in dollars and q is the...Ch. 11.5 - In Problems 1 -8, p is in dollars and q is the...Ch. 11.5 - In Problems 1 -8, p is in dollars and q is the...Ch. 11.5 - In Problems 1 -8, p is in dollars and q is the...Ch. 11.5 - In Problems 1 -8, p is in dollars and q is the...Ch. 11.5 - In Problems 1 -8, p is in dollars and q is the...Ch. 11.5 - In Problems 1 -8, p is in dollars and q is the...Ch. 11.5 - 9. Suppose the demand function for a product is...Ch. 11.5 - 10. Suppose the weekly demand function for a...Ch. 11.5 - In Problems 11 and 12, the demand functions for...Ch. 11.5 - In Problems 11 and 12, the demand functions for...Ch. 11.5 - In Problems 15-24, p is the price per unit in...Ch. 11.5 - In Problems 15-24, p is the price per unit in...Ch. 11.5 - In Problems 15-24, p is the price per unit in...Ch. 11.5 - In Problems 15-24, p is the price per unit in...Ch. 11.5 - In Problems 15-24, p is the price per unit in...Ch. 11.5 - In Problems 15-24, p is the price per unit in...Ch. 11.5 - In Problems 15-24, p is the price per unit in...Ch. 11.5 - In Problems 15-24, p is the price per unit in...Ch. 11.5 - In Problems 15-24, p is the price per unit in...Ch. 11.5 - In Problems 15-24, p is the price per unit in...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - In Problems 1-12, find the derivative of each...Ch. 11 - 13. Write the equation of the line tangent to
Ch. 11 - 14. At write the equation of the line tangent to...Ch. 11 - In Problems 15-20, find the indicated...Ch. 11 - In Problems 15-20, find the indicated...Ch. 11 - In Problems 15-20, find the indicated...Ch. 11 - In Problems 15-20, find the indicated...Ch. 11 - In Problems 15-20, find the indicated...Ch. 11 - In Problems 15-20, find the indicated...Ch. 11 - 21. Find the slope of the tangent to the curve
Ch. 11 - 22. Find the points where tangents to the graph of...Ch. 11 - 23. Suppose where x and y are differentiable...Ch. 11 - 24. A right triangle with legs of lengths x and y...Ch. 11 - 25. Demographics Using U.S. Census Bureau data for...Ch. 11 - 26. Life expectancy at age 65 Using Social...Ch. 11 - 27. Compound interest If $1000 is invested for n...Ch. 11 - 28. Disposable income Disposable income is the...Ch. 11 - 29. Radioactive decay A breeder reactor converts...Ch. 11 - 30. Marginal cost The average cost of producing x...Ch. 11 - 31. Inflation The impact of inflation on a $60,000...Ch. 11 - 32. Evaporation A spherical droplet of water...Ch. 11 - 33. Worker safety A sign is being lowered over the...Ch. 11 - 34. Environment Suppose that in a study of water...Ch. 11 - 35. Elasticity A demand function is given...Ch. 11 - 36. Elasticity Suppose the demand for a product is...Ch. 11 - 37. Elasticity Suppose the weekly demand function...Ch. 11 - 38. Revenue A product has the demand...Ch. 11 - 39. Taxes Can increasing the tax per unit sold...Ch. 11 - 40. Taxes Suppose the demand and supply functions...Ch. 11 - 41. Taxes Suppose the supply and demand functions...Ch. 11 - In Problems 1-8, find the derivative of each...Ch. 11 - In Problems 1-8, find the derivative of each...Ch. 11 - In Problems 1-8, find the derivative of each...Ch. 11 - In Problems 1-8, find the derivative of each...Ch. 11 - In Problems 1-8, find the derivative of each...Ch. 11 - In Problems 1-8, find the derivative of each...Ch. 11 - In Problems 1-8, find the derivative of each...Ch. 11 - In Problems 1-8, find the derivative of each...Ch. 11 -
9. Find y’ if .
Ch. 11 - 10. Let
If , find when x = 6 and y = 8.
Ch. 11 - 11. Find y’ if .
Ch. 11 - 12. Suppose the weekly revenue and weekly cost...Ch. 11 - 13. Suppose the demand for a product is , where p...Ch. 11 - 14. Suppose the demand function for a product is...Ch. 11 - 15. The sales of a product are given by , where S...Ch. 11 - 16. Using U.S. Centers for Medicare and Medicaid...Ch. 11 - 17. Suppose the demand and supply functions for a...Ch. 11 - 19. Prices for goods and services tend to rise...

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