Explanation Given: The equation of the curve is y 3 = 8 x 4 . Also, the point is ( 1 , 2 ) . Formula used: Chain Rule: If f and g are functions and y = [ g ( x ) ] , then y ′ = f ′ [ g ( x ) ] ⋅ g ′ ( x ) provided that f ′ [ g ( x ) ] and g ′ ( x ) exists. And, The equation of tangent line is given as: y − y 1 = m ( x − x 1 ) Here, m is the slope, x 1 and y 1 are the given points. The equation for slope is: m = d y d x . Calculation: Implicit Differentiation: This differentiation is used to evaluate d y d x , when y is defined implicitly as a function of x . The differentiation is carried out in two steps: Step 1: Both the sides of the given equation are differentiated with respect to x . Since y is the function of x , always use the chain rule. d d x ( y 3 ) = d d x ( 8 x 4 ) 3 y 2 d y d x = 8 d d x ( x 4 ) 3 y 2 d y d x = 8 ( 4 x 3 ) 3 y 2 d y d x = 32 x 3 Step 2: Solve the resultant equation for d y d x

Mathematics with Applications In the Management, Natural, and Social Sciences Plus NEW MyLab Math with Pearson eText -- Access Card Package (11th Edition)

11th Edition

ISBN: 9780321935441

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 12.4, Problem 31E

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To calculate: The equation of the tangent line at the given point (1,2) for the given curve, y3=8x4.

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Chapter 12.4, Problem 30E

Chapter 12.4, Problem 32E

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Chapter 12 Solutions

Mathematics with Applications In the Management, Natural, and Social Sciences Plus NEW MyLab Math with Pearson eText -- Access Card Package (11th Edition)

Ch. 12.1 - Checkpoint 1 For what values of x is the function...Ch. 12.1 - Checkpoint 2 Find all intervals on which is...Ch. 12.1 - Checkpoint 3 Identity the x-values of all points...Ch. 12.1 - Checkpoint 4 Find the critical numbers for each of...Ch. 12.1 - Prob. 5CP Ch. 12.1 - Prob. 6CP Ch. 12.1 - Prob. 7CP Ch. 12.1 - Prob. 8CP Ch. 12.1 - Checkpoint 9 If a sales function is given by...Ch. 12.1 - Prob. 1E

Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E Ch. 12.1 - Prob. 5E Ch. 12.1 - Prob. 6E Ch. 12.1 - Prob. 7E Ch. 12.1 - Prob. 8E Ch. 12.1 - Prob. 9E Ch. 12.1 - Prob. 10E Ch. 12.1 - Prob. 11E Ch. 12.1 - Find the intervals on which each function is...Ch. 12.1 - Find the intervals on which each function is...Ch. 12.1 - Find the intervals on which each function is...Ch. 12.1 - Prob. 15E Ch. 12.1 - Prob. 16E Ch. 12.1 - Prob. 17E Ch. 12.1 - Prob. 18E Ch. 12.1 - Prob. 19E Ch. 12.1 - Prob. 20E Ch. 12.1 - Prob. 21E Ch. 12.1 - Determine the location of each local extremum of...Ch. 12.1 - Prob. 23E Ch. 12.1 - Prob. 24E Ch. 12.1 - Prob. 25E Ch. 12.1 - Prob. 26E Ch. 12.1 - Determine the location of each local extremum of...Ch. 12.1 - Determine the location of each local extremum of...Ch. 12.1 - In Exercises 29–40, use the first-derivative test...Ch. 12.1 - In Exercises 29–40, use the first-derivative test...Ch. 12.1 - In Exercises 29–40, use the first-derivative test...Ch. 12.1 - In Exercises 29–40, use the first-derivative test...Ch. 12.1 - In Exercises 29–40, use the first-derivative test...Ch. 12.1 - In Exercises 29–40, use the first-derivative test...Ch. 12.1 - In Exercises 29–40, use the first-derivative test...Ch. 12.1 - Prob. 36E Ch. 12.1 - Prob. 37E Ch. 12.1 - Prob. 38E Ch. 12.1 - Prob. 39E Ch. 12.1 - Prob. 40E Ch. 12.1 - Use the maximum/minimum finder on a graphing...Ch. 12.1 - Prob. 42E Ch. 12.1 - Prob. 43E Ch. 12.1 - Prob. 44E Ch. 12.1 - Prob. 45E Ch. 12.1 - 46. Business Operating profits (in billions of...Ch. 12.1 - Prob. 47E Ch. 12.1 - Prob. 48E Ch. 12.1 - Prob. 49E Ch. 12.1 - Prob. 50E Ch. 12.1 - 51. Physical Science A Boston Red Sox pitcher...Ch. 12.1 - 52. Business Revenue (in billions of dollars) from...Ch. 12.1 - 53. Social Science According to projections by the...Ch. 12.1 - 54. Business The number of full-time employees (in...Ch. 12.1 - Work these exercises. You may need to use the...Ch. 12.1 - 56. Business The number of common stock holders...Ch. 12.1 - 57. Natural Science The microbe concentration...Ch. 12.1 - Prob. 58E Ch. 12.1 - Prob. 59E Ch. 12.1 - Prob. 60E Ch. 12.1 - Prob. 61E Ch. 12.1 - Prob. 62E Ch. 12.1 - Prob. 63E Ch. 12.1 - Prob. 64E Ch. 12.1 - 65. Social Science A group of researchers found...Ch. 12.1 - Prob. 66E Ch. 12.1 - Prob. 67E Ch. 12.1 - Prob. 68E Ch. 12.1 - Prob. 69E Ch. 12.1 - Prob. 70E Ch. 12.2 - Prob. 1CP Ch. 12.2 - Prob. 2CP Ch. 12.2 - Prob. 3CP Ch. 12.2 - Prob. 4CP Ch. 12.2 - Prob. 5CP Ch. 12.2 - Prob. 6CP Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - For each of these functions, find and (See...Ch. 12.2 - In Exercises 19 and 20, P(t) is the price of a...Ch. 12.2 - In Exercise 19 and 20, is the price of a certain...Ch. 12.2 - Physical Science Each of the functions in...Ch. 12.2 - Physical Science Each of the functions in...Ch. 12.2 - Prob. 23E Ch. 12.2 - Prob. 24E Ch. 12.2 - Prob. 25E Ch. 12.2 - Prob. 26E Ch. 12.2 - Find the largest open intervals on which each...Ch. 12.2 - Prob. 28E Ch. 12.2 - Find the largest open intervals on which each...Ch. 12.2 - Find the largest open intervals on which each...Ch. 12.2 - Find the largest open intervals on which each...Ch. 12.2 - Find the largest open intervals on which each...Ch. 12.2 - Business In Exercises 33–36, find the point of...Ch. 12.2 - Business In Exercises 33–36, find the point of...Ch. 12.2 - Business In Exercises 33–36, find the point of...Ch. 12.2 - Business In Exercises 33–36, find the point of...Ch. 12.2 - Find all critical numbers of the functions in...Ch. 12.2 - Find all critical numbers of the functions in...Ch. 12.2 - Find all critical numbers of the functions in...Ch. 12.2 - Prob. 40E Ch. 12.2 - Prob. 41E Ch. 12.2 - Prob. 42E Ch. 12.2 - Prob. 43E Ch. 12.2 - Prob. 44E Ch. 12.2 - Prob. 45E Ch. 12.2 - Find all critical numbers of the functions in...Ch. 12.2 - Prob. 47E Ch. 12.2 - Prob. 48E Ch. 12.2 - Prob. 49E Ch. 12.2 - Prob. 50E Ch. 12.2 - Prob. 51E Ch. 12.2 - Prob. 52E Ch. 12.2 - Prob. 53E Ch. 12.2 - Prob. 54E Ch. 12.2 - Prob. 55E Ch. 12.2 - Prob. 56E Ch. 12.2 - Prob. 57E Ch. 12.2 - Prob. 58E Ch. 12.2 - Prob. 59E Ch. 12.2 - Prob. 60E Ch. 12.2 - Prob. 61E Ch. 12.2 - Prob. 62E Ch. 12.2 - Prob. 63E Ch. 12.2 - 64. Business The price of Starbucks Corporation...Ch. 12.2 - 65. Social Science The population of Wyoming (in...Ch. 12.2 - Prob. 66E Ch. 12.2 - Prob. 67E Ch. 12.2 - Prob. 68E Ch. 12.2 - Prob. 69E Ch. 12.3 - Checkpoint 1 Find the location of the absolute...Ch. 12.3 - Prob. 2CP Ch. 12.3 - Prob. 3CP Ch. 12.3 - Prob. 4CP Ch. 12.3 - Prob. 5CP Ch. 12.3 - Checkpoint 6 In Example 9, suppose annual demand...Ch. 12.3 - Find the location of the absolute maximum and...Ch. 12.3 - Find the location of the absolute maximum and...Ch. 12.3 - Find the location of the absolute maximum and...Ch. 12.3 - Find the location of the absolute maximum and...Ch. 12.3 - Find the location of the absolute maximum and...Ch. 12.3 - Find the location of the absolute maximum and...Ch. 12.3 - Find the absolute extrema of each function on the...Ch. 12.3 - Find the absolute extrema of each function on the...Ch. 12.3 - Find the absolute extrema of each function on the...Ch. 12.3 - Find the absolute extrema of each function on the...Ch. 12.3 - Prob. 11E Ch. 12.3 - Find the absolute extrema of each function on the...Ch. 12.3 - Find the absolute extrema of each function on the...Ch. 12.3 - Find the absolute extrema of each function on the...Ch. 12.3 - Find the absolute extrema of each function on the...Ch. 12.3 - Prob. 16E Ch. 12.3 - Prob. 17E Ch. 12.3 - Prob. 18E Ch. 12.3 - Prob. 19E Ch. 12.3 - If possible, find an absolute extremum of each...Ch. 12.3 - If possible, find an absolute extremum of each...Ch. 12.3 - If possible, find an absolute extremum of each...Ch. 12.3 - If possible, find an absolute extremum of each...Ch. 12.3 - Prob. 24E Ch. 12.3 - Work these problems. (See Example 5.) 25. Business...Ch. 12.3 - Work these problems. (See Example 5.) 26. Business...Ch. 12.3 - 27. Business The number of employees of...Ch. 12.3 - 28. Social Science The number of people in...Ch. 12.3 - 29. Business Revenue generated by admissions to...Ch. 12.3 - 30. Health The number of heart transplants can be...Ch. 12.3 - 31. Business A manufacturer produces gas grills...Ch. 12.3 - 32. Business Saltwater taffy can be sold wholesale...Ch. 12.3 - 33. Natural Science A lake polluted by bacteria is...Ch. 12.3 - 34. Social Science The number of people in...Ch. 12.3 - 35. Health A disease has hit College Station,...Ch. 12.3 - Prob. 36E Ch. 12.3 - Prob. 37E Ch. 12.3 - Prob. 38E Ch. 12.3 - Prob. 39E Ch. 12.3 - Prob. 40E Ch. 12.3 - Prob. 41E Ch. 12.3 - 42. Business A cylindrical can of volume 58 cubic...Ch. 12.3 - Prob. 43E Ch. 12.3 - Prob. 44E Ch. 12.3 - Prob. 45E Ch. 12.3 - 46. Business A rectangular field is to be enclosed...Ch. 12.3 - 47. Business A mathematics book is to contain 36...Ch. 12.3 - Prob. 48E Ch. 12.3 - 49. Business If the price charged for a candy bar...Ch. 12.3 - 50. Business A company makes plastic buckets for...Ch. 12.3 - 51. Business We can use the function to model the...Ch. 12.3 - 52. Business A rock-and-roll band travels from...Ch. 12.3 - 53. Natural Science Homing pigeons avoid flying...Ch. 12.3 - 54. Business A company wishes to run a utility...Ch. 12.3 - Prob. 55E Ch. 12.3 - Prob. 56E Ch. 12.3 - Prob. 57E Ch. 12.3 - Prob. 58E Ch. 12.3 - Prob. 59E Ch. 12.3 - 60. Business A restaurant has an annual demand for...Ch. 12.3 - 61. Business Choose the correct answer:* The...Ch. 12.4 - Checkpoint 1 Find for Ch. 12.4 - Prob. 2CP Ch. 12.4 - Prob. 3CP Ch. 12.4 - Prob. 4CP Ch. 12.4 - Prob. 5CP Ch. 12.4 - Prob. 6CP Ch. 12.4 - Checkpoint 7 Suppose the sales function in Example...Ch. 12.4 - Prob. 1E Ch. 12.4 - Prob. 2E Ch. 12.4 - Find by implicit differentiation. (See Examples...Ch. 12.4 - Find by implicit differentiation. (See Examples...Ch. 12.4 - Prob. 5E Ch. 12.4 - Prob. 6E Ch. 12.4 - Prob. 7E Ch. 12.4 - Prob. 8E Ch. 12.4 - Prob. 9E Ch. 12.4 - Prob. 10E Ch. 12.4 - Prob. 11E Ch. 12.4 - Find by implicit differentiation. (See Examples...Ch. 12.4 - Prob. 13E Ch. 12.4 - Prob. 14E Ch. 12.4 - Prob. 15E Ch. 12.4 - Find by implicit differentiation. (See Examples...Ch. 12.4 - Prob. 17E Ch. 12.4 - Prob. 18E Ch. 12.4 - Prob. 19E Ch. 12.4 - Find at the given point. (See Example 5.) 20. Ch. 12.4 - Find at the given point. (See Example 5.) 21. Ch. 12.4 - Prob. 22E Ch. 12.4 - Prob. 23E Ch. 12.4 - Find at the given point. (See Example 5.) 23. 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Business At a certain online printing service,...Ch. 12.5 - Checkpoint 1 Given that R3 = 25n4, find when n =...Ch. 12.5 - Prob. 2CP Ch. 12.5 - Prob. 3CP Ch. 12.5 - Prob. 4CP Ch. 12.5 - Prob. 5CP Ch. 12.5 - Prob. 6CP Ch. 12.5 - Prob. 7CP Ch. 12.5 - Given that x and y are functions of time, find the...Ch. 12.5 - Given that x and y are functions of time, find the...Ch. 12.5 - Given that x and y are functions of time, find the...Ch. 12.5 - Given that x and y are functions of time, find the...Ch. 12.5 - Prob. 5E Ch. 12.5 - Prob. 6E Ch. 12.5 - Prob. 7E Ch. 12.5 - Prob. 8E Ch. 12.5 - Prob. 9E Ch. 12.5 - Given that x and y are functions of time, find the...Ch. 12.5 - Work these exercises. (See Examples 1, 3, and...Ch. 12.5 - Work these exercises. (See Examples 1, 3, and...Ch. 12.5 - Work these exercises. (See Examples 1, 3, and...Ch. 12.5 - Work these exercises. (See Examples 1, 3, and...Ch. 12.5 - Work these exercises. (See Examples 1, 3, and...Ch. 12.5 - Work these exercises. (See Examples 1, 3, and...Ch. 12.5 - Prob. 17E Ch. 12.5 - Prob. 18E Ch. 12.5 - Prob. 19E Ch. 12.5 - Work these exercises. (See Examples 1, 3, and...Ch. 12.5 - 21. Business An architectural firm must decide on...Ch. 12.5 - 22. Social Science During a six-game hitless slump...Ch. 12.5 - Work these exercises. (See Example...Ch. 12.5 - Work these exercises. (See Example...Ch. 12.5 - Work these exercises. (See Example...Ch. 12.5 - Work these exercises. (See Example...Ch. 12.5 - Work these exercises. 27. Business The campus...Ch. 12.5 - Work these exercises. 28. Business Following a...Ch. 12.5 - 29. Business During a local political race, the...Ch. 12.5 - Prob. 30E Ch. 12.5 - Prob. 31E Ch. 12.5 - Prob. 32E Ch. 12.6 - Prob. 1CP Ch. 12.6 - Prob. 2CP Ch. 12.6 - Prob. 3CP Ch. 12.6 - Prob. 4CP Ch. 12.6 - Prob. 1E Ch. 12.6 - Sketch the graph of the function. Identify any...Ch. 12.6 - Prob. 3E Ch. 12.6 - Prob. 4E Ch. 12.6 - Sketch the graph of the function. Identify any...Ch. 12.6 - Prob. 6E Ch. 12.6 - Sketch the graph of the function. Identify any...Ch. 12.6 - Prob. 8E Ch. 12.6 - Prob. 9E Ch. 12.6 - Prob. 10E Ch. 12.6 - Prob. 11E Ch. 12.6 - Sketch the graph of the function. Identify any...Ch. 12.6 - Prob. 13E Ch. 12.6 - Prob. 14E Ch. 12.6 - Prob. 15E Ch. 12.6 - Prob. 16E Ch. 12.6 - Prob. 17E Ch. 12.6 - Sketch the graph of the function. Identify any...Ch. 12.6 - Prob. 19E Ch. 12.6 - Prob. 20E Ch. 12.6 - Prob. 21E Ch. 12.6 - Prob. 22E Ch. 12.6 - Prob. 23E Ch. 12.6 - In Exercises 23–28, sketch the graph of a function...Ch. 12.6 - Prob. 25E Ch. 12.6 - In Exercises 23–28, sketch the graph of a function...Ch. 12.6 - In Exercises 23–28, sketch the graph of a function...Ch. 12.6 - In Exercises 23–28, sketch the graph of a function...Ch. 12.6 - 29. Business The accompanying figure shows the...Ch. 12.6 - 30. Refer to the graph in Exercise 29. Which...Ch. 12.6 - 31. Business The revenue function (in millions of...Ch. 12.6 - 32. Business The cost function (in millions of...Ch. 12.6 - 33. Social Science The percent of the population...Ch. 12.6 - 34. Health The percent of female graduates in...Ch. 12.6 - Use a graphing calculator or computer to find the...Ch. 12.6 - Use a graphing calculator or computer to find the...Ch. 12.6 - Use a graphing calculator or computer to find the...Ch. 12.6 - Use a graphing calculator or computer to find the...Ch. 12.6 - 39. Business The revenue (in billions of dollars)...Ch. 12.6 - 40. Business The revenue (in billions of dollars)...Ch. 12.6 - Use a graphing calculator or computer to find the...Ch. 12.6 - Prob. 42E Ch. 12.6 - Prob. 43E Ch. 12.6 - Prob. 44E Ch. 12 - Prob. 1CE Ch. 12 - Prob. 2CE Ch. 12 - Prob. 3CE Ch. 12 - Prob. 4CE Ch. 12 - Prob. 5CE Ch. 12 - 6. 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The equation of the tangent line at the given point ( 1 , 2 ) for the given curve, y 3 = 8 x 4 .

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Chapter 12 Solutions