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12th Edition

Ronald J. Harshbarger + 1 other

Publisher: Cengage Learning

ISBN: 9781337625340

Chapter 11.3, Problem 48E

Textbook Problem

5 views

At what points does the curve defined by

(a) horizontal tangents?

(b) vertical tangents?

**(a)**

To determine

**To calculate:** The point at which the curve defined by

**Given Information:**

The provided equation of the curve is,

**Formula used:**

The slope of tangent to a curve

The power rule of differentiation:

**Calculation:**

Consider the provided equation of curve,

Now, find the derivative

**(b)**

To determine

**To calculate:** The point at which the curve defined by

Mathematical Applications for the Management, Life, and Social Sciences

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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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