Solved: Point of Diminishing Returns In Exercise, find the

Problem 4E Find f?(x) for the function. Then find f?(0) and f?(2).

Problem 95A Velocity and Acceleration A car is moving along a straight stretch of road. The acceleration of the car is given by the graph shown. Assume that the velocity of the car is always positive. At what time was the car moving most rapidly? Explain. Source: Larry Taylor.

Problem 5E Find f?(x) for the function. Then find f?(0) and f?(2). f(x) = 3x2 ? 4x + 8

Problem 2E Find f?(x) for the function. Then find f?(0) and f?(2). f(x) = 4x3 + 5x2 + 6x ? 7

Problem 1E Find f?(x) for the function. Then find f?(0) and f?(2). f(x) = 5x3? 7x2 + 4x + 3

Problem 6E Find f?(x) for the function. Then find f?(0) and f?(2). f(x) = 8x2 + 6x + 5

Problem 7E Find f?(x) for the function. Then find f?(0) and f?(2).

Problem 8E Find f?(x) for the function. Then find f?(0) and f?(2).

Problem 9E Find f?(x) for the function. Then find f?(0) and f?(2).

Problem 10E Find f?(x) for the function. Then find f?(0) and f?(2).

Problem 11E Find f?(x) for the function. Then find f?(0) and f?(2). f(x) = 32x3/4

Problem 12E Find f?(x) for the function. Then find f?(0) and f?(2). f(x) = ?6x1/3

Problem 14E Find f?(x) for the function. Then find f?(0) and f?(2).

Problem 13E Find f?(x) for the function. Then find f?(0) and f?(2).

Problem 16E Find f?(x) for the function. Then find f?(0) and f?(2).

Problem 15E Find f?(x) for the function. Then find f?(0) and f?(2).

Problem 17E Find f?(x), the third derivative of f, and f?(x), the fourth derivative of f, for the function. f(x) = 7x4 + 6x3 + 5x2 + 4x + 3

Problem 18E Find f?(x), the third derivative of f, and f?(x), the fourth derivative of f, for the function. f(x) = ?2x4 + 7x3 + 4x2 + x

Problem 19E Find f?(x), the third derivative of f, and f?(x), the fourth derivative of f, for the function. f(x) = 5x5 ? 3x4 + 2x3 + 7x2 + 4

Problem 20E Find f?(x), the third derivative of f, and f?(x), the fourth derivative of f, for the function. f(x) = 2x5 + 3x4 ? 5x3 + 9x ? 2

Problem 21E Find f?(x), the third derivative of f, and f?(x), the fourth derivative of f, for the function.

Problem 22E Find f?(x), the third derivative of f, and f?(x), the fourth derivative of f, for the function.

Problem 23E Find f?(x), the third derivative of f, and f?(x), the fourth derivative of f, for the function.

Problem 24E Find f?(x), the third derivative of f, and f?(x), the fourth derivative of f, for the function.

Problem 25E Let f(x) = ln x. a. Compute f?(x), f?(x), f??(x), f(4)(x), and f(5)(x). ________________ b. Guess a formula for f(n)(x), where n is any positive integer.

Problem 27E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 26E For f(x) = ex, find f?(x) and f??(x). What is the nth derivative of f with respect to x?

Problem 28E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 29E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 30E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 31E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 32E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 33E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = x2 + 10x ? 9 Step-by-step solution

Problem 35E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = ?2x3 + 9x2 + 168x ? 3

Problem 34E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = 8 ? 6x ? x2

Problem 36E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = –x3 ? 12x2 ? 45x + 2

Problem 37E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 38E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 40E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = –x(x ? 3)2

Problem 41E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = 18x ? 18e?x

Problem 39E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = x(x + 5)2

Problem 42E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 43E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = x8/3 ? 4x5/3

Problem 44E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = x7/3 + 56x4/3

Problem 46E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = x2 + 8 ln |x + 1|

Problem 45E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = ln(x2 + 1)

Problem 47E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = x2 log |x|

Problem 48E In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 49E For the of the exercise listed below, suppose that the function that is graphed is not f(x),but f?(x). Find the open intervals where the original function is concave upward or concave downward, and find the location of any inflection points. Exercise In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 50E For the of the exercise listed below, suppose that the function that is graphed is not f(x),but f?(x). Find the open intervals where the original function is concave upward or concave downward, and find the location of any inflection points. Exercise In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 52E For the of the exercise listed below, suppose that the function that is graphed is not f(x),but f?(x). Find the open intervals where the original function is concave upward or concave downward, and find the location of any inflection points. Exercise In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 51E For the of the exercise listed below, suppose that the function that is graphed is not f(x),but f?(x). Find the open intervals where the original function is concave upward or concave downward, and find the location of any inflection points. Exercise In Exercise find the open intervals where the functions are concave upward or concave downward. Find any inflection points.

Problem 53E Give an example of a function f(x) such that f?(0) = 0 but f?(0) does not exist. Is there a relative minimum or maximum or an inflection point at x = 0?

Problem 54E a. Graph the two functions f(x) = x7/3 and g(x) = x5/3 on the window [?2, 2] by [2, 2]. ________________ b. Verify that both f and g have an inflection point at (0, 0). ________________ c. How is the value of f?(0) different from g?(0)? ________________ d. Based on what you have seen so far in this exercise, is it always possible to tell the difference between a point where the second derivative is 0 or undefined based on the graph? Explain.

Problem 55E Describe the slope of the tangent line to the graph of f(x) = ex for the following. a.x ? ?? ________________ b.x ? 0

Problem 56E What is true about the slope of the tangent line to the graph of f(x) = ln x as x ? ?? As x ? 0?

Problem 57E Find any critical numbers for f in Exercise and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f?(c) = 0 or f?(c) does not exist for a critical number c, then the second derivative test gives no information. In this case, use the first derivative test instead. f(x) = –x2? 10x ? 25

Problem 58E Find any critical numbers for f in Exercise and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f?(c) = 0 or f?(c) does not exist for a critical number c, then the second derivative test gives no information. In this case, use the first derivative test instead. f(x) = x2? 12x + 36

Problem 59E Find any critical numbers for f in Exercise and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f?(c) = 0 or f?(c) does not exist for a critical number c, then the second derivative test gives no information. In this case, use the first derivative test instead. f(x) = 3x3 ? 3x2 + 1

Problem 60E Find any critical numbers for f in Exercise and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f?(c) = 0 or f?(c) does not exist for a critical number c, then the second derivative test gives no information. In this case, use the first derivative test instead. f(x) = 2x3 ? 4x2 + 2

Problem 61E Find any critical numbers for f in Exercise and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f?(c) = 0 or f?(c) does not exist for a critical number c, then the second derivative test gives no information. In this case, use the first derivative test instead. f(x) = (x + 3)4

Problem 62E Find any critical numbers for f in Exercise and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f?(c) = 0 or f?(c) does not exist for a critical number c, then the second derivative test gives no information. In this case, use the first derivative test instead. f(x) = x3

Problem 63E Find any critical numbers for f in Exercise and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f?(c) = 0 or f?(c) does not exist for a critical number c, then the second derivative test gives no information. In this case, use the first derivative test instead. f(x) = x7/3 + x4/3

Problem 64E Find any critical numbers for f in Exercise and then use the second derivative test to decide whether the critical numbers lead to relative maxima or relative minima. If f?(c) = 0 or f?(c) does not exist for a critical number c, then the second derivative test gives no information. In this case, use the first derivative test instead. f(x) = x8/3 + x5/3

Problem 65E Sometimes the derivative of a function is known, but not the function. We will see more of this later in the book. For each function f?defined in Exercise find f?(x), then use a graphing calculator to graph f?and f?in the indicated window. Use the graph to do the following. a. Give the (approximate)x-values where f has a maximum or minimum. ________________ b. By considering the sign of f?(x), give the (approximate) intervals where f(x) is increasing and decreasing. ________________ c. Give the (approximate)x-values of any inflection points. ________________ d. By considering the sign of f?(x), give the intervals where f is concave upward or concave downward. f?(x) = x3 ? 6x2 + 7x + 4; [?5, 5] by [?5, 15]

Problem 66E Sometimes the derivative of a function is known, but not the function. We will see more of this later in the book. For each function f? defined in Exercise find f?(x), then use a graphing calculator to graph f?and f?in the indicated window. Use the graph to do the following. a. Give the (approximate)x-values where f has a maximum or minimum. ________________ b. By considering the sign of f?(x), give the (approximate) intervals where f(x) is increasing and decreasing. ________________ c. Give the (approximate)x-values of any inflection points. ________________ d. By considering the sign of f?(x), give the intervals where f is concave upward or concave downward. f?(x) = 10x2(x ? 1)(5x ? 3); [?1, 1.5] by [?20, 20]

Problem 67E Sometimes the derivative of a function is known, but not the function. We will see more of this later in the book. For each function f?defined in Exercise find f?(x), then use a graphing calculator to graph f?and f?in the indicated window. Use the graph to do the following. a. Give the (approximate)x-values where f has a maximum or minimum. ________________ b. By considering the sign of f?(x), give the (approximate) intervals where f(x) is increasing and decreasing. ________________ c. Give the (approximate)x-values of any inflection points. ________________ d. By considering the sign of f?(x), give the intervals where f is concave upward or concave downward.

Problem 68E Sometimes the derivative of a function is known, but not the function. We will see more of this later in the book. For each function f?defined in Exercise find f?(x), then use a graphing calculator to graph f?and f?in the indicated window. Use the graph to do the following. a. Give the (approximate)x-values where f has a maximum or minimum. ________________ b. By considering the sign of f?(x), give the (approximate) intervals where f(x) is increasing and decreasing. ________________ c. Give the (approximate)x-values of any inflection points. ________________ d. By considering the sign of f?(x), give the intervals where f is concave upward or concave downward. f?(x) = x2 + x ln x; [0, 1] by [?2,2]

Problem 69E Suppose a friend makes the following argument. A function f is increasing and concave downward. Therefore, f? is positive and decreasing, so it eventually becomes 0 and then negative, at which point f decreases. Show that your friend is wrong by giving an example of a function that is always increasing and concave downward.

Problem 70A Product Life Cycle The accompanying figure shows the product life cycle graph, with typical products marked on it. It illustrates the fact that a new product is often purchased at a faster and faster rate as people become familiar with it. In time, saturation is reached and the purchase rate stays constant until the product is made obsolete by newer products, after which it is purchased less and less. Source: tutor2u. a. Which products on the left side of the graph are closest to the left-hand inflection point? What does the inflection point mean here? ________________ b. Which product on the right side of the graph is closest to the right-hand inflection point? What does the inflection point mean here? ________________ c. Discuss where portable Blu-ray players, iPads, and other new technologies should be placed on the graph.

Problem 72A Point of Diminishing Returns In Exercise, find the point of diminishing returns (x, y) for the given functions, where R(x), represents revenue (in thousands of dollars) and xrepresents the amount spent on advertising (in thousands of dollars). R(x) = 10,000 ? x3 + 42x2 + 800x, 0 ?x ? 20

Problem 71A Social Security Assets As seen in the first section of this chapter, the projected year-end assets in the Social Security trust funds, in trillions of dollars, where t represents the number of years since 2000, can be approximated by A(t) = 0.0000329t3 ? 0.00450t2 + 0.0613t + 2.34, where 0 ? t ? 50. Find the value of t when Social Security assets will decrease most rapidly. Approximately when does this occur? Source: Social Security Administration.

Problem 73A Point of Diminishing Returns In Exercise, find the point of diminishing returns (x, y) for the given functions, where R(x), represents revenue (in thousands of dollars) and xrepresents the amount spent on advertising (in thousands of dollars).

Problem 74A Point of Diminishing Returns In Exercise, find the point of diminishing returns (x, y) for the given functions, where R(x), represents revenue (in thousands of dollars) and xrepresents the amount spent on advertising (in thousands of dollars). R(x) = ?0.3x3 + x2 + 11.4x, 0 ?x ? 6

Problem 75A Point of Diminishing Returns In Exercise, find the point of diminishing returns (x, y) for the given functions, where R(x), represents revenue (in thousands of dollars) and xrepresents the amount spent on advertising (in thousands of dollars). R(x) = ?0.6x3 + 3.7x2 + 5x, 0 ?x ? 6

Problem 76A Risk Aversion In economics, an index of absolute risk aversion is defined as where M measures how much of a commodity is owned and U(M) is a utility function, which measures the ability of quantity M of a commodity to satisfy a consumer’s wants. Find I(M) for and for U(M) = M2/3, and determine which indicates a greater aversion to risk.

Problem 77A Demand Function The authors of an article in an economics journal state that if D(q) is the demand function, then the inequality qD?(q)+ D?(q) < 0 is equivalent to saying that the marginal revenue declines more quickly than does the price. Prove that this equivalence is true. Source: Bell Journal of Economics.

Problem 78A Population Growth When a hardy new species is introduced into an area, the population often increases as shown. Explain the significance of the following function values on the graph. a.f0 ________________ b.f(a) ________________ c.fM

Problem 79A Bacteria Population Assume that the number of bacteria R(t) (in millions) present in a certain culture at time t (in hours) is given by R(t) = t2(t ? 18) + 96t + 1000. a. At what time before 8 hours will the population be maximized? ________________ b. Find the maximum population.

Problem 80A Ozone Depletion According to an article in The New York Times, “Government scientists reported last week that they had detected a slowdown in the rate at which chemicals that deplete the earth’s protective ozone layer are accumulating in the atmosphere.” Letting c(t) be the amount of ozone-depleting chemicals at time t, what does this statement tell you about c(t), c?(t), and c?(t)? Source: The New York Times.

Problem 81A Drug Concentration The percent of concentration of a certain drug in the bloodstream x hours after the drug is administered is given by . For example, after 1 hour the concentration is given by a. Find the time at which concentration is a maximum. ________________ b. Find the maximum concentration.

Problem 82A Drug Concentration The percent of concentration of a drug in the bloodstream x hours after the drug is administered is given by . a. Find the time at which the concentration is a maximum. ________________ b. Find the maximum concentration.

Problem 83A The exercise are a continuation of exercise first given in the section on Derivatives of Exponential Functions. Find the inflection point of the graph of the logistic function. This is the point at which the growth rate begins to decline. Insect Growth The growth function for a population of beetles is given by

Problem 84A The exercise are a continuation of exercise first given in the section on Derivatives of Exponential Functions. Find the inflection point of the graph of the logistic function. This is the point at which the growth rate begins to decline. Clam Population Growth The population of a bed of clams is described by .

Problem 85A Clam Growth Researchers used a version of the Gompertz curve to model the growth of razor clams during the first seven years of the clams’ lives with the equation . where L(t) gives the length (in centimeters) after t years, B = 14.3032, c = 7.267963, and k = 0.670840. Find the inflection point and describe what it signifies. Source: Journal of Experimental Biology. Hints for Exercise: Leave B, c, and k as constants until you are ready to calculate your final answer.

Problem 86A Breast Cancer Growth Researchers used a version of the Gompertz curve to model the growth of breast cancer tumors with the equation where N(t) is the number of cancer cells after t days, c = 27.3, and k = 0.011. Find the inflection point and describe what it signifies. Source: Cancer Research. Hints for Exercise: Leave B, c, and k as constants until you are ready to calculate your final answer.

Problem 87A Popcorn Researchers have determined that the amount of moisture present in a kernel of popcorn affects the volume of the popped corn and can be modeled for certain sizes of kernels by the function v(x) = –35.98 + 12.09x ? 0.4450x2, where x is moisture content (%, wet basis) and v(x) is the expansion volume (in cm3/gram). Describe the concavity of this function. Source: Cereal Chemistry.

Problem 89A Crime In 1995, the rate of violent crimes in New York City continued to decrease, but at a slower rate than in previous years. Letting f(t) be the rate of violent crime as a function of time, what does this tell you about f(t), f?(t), and f?(t)? Source: The New York Times.

Problem 88A Alligator Teeth Researchers have developed a mathematical model that can be used to estimate the number of teeth N(t) at time t (days of incubation) for Alligator mississippiensis,where Find the inflection point and describe its importance to this research. Source: Journal of Theoretical Biology.

Problem 90A Chemical Reaction An autocatalytic chemical reaction is one in which the product being formed causes the rate of formation to increase. The rate of a certain autocatalytic reaction is given by V(x) = 12x(100?x), where x is the quantity of the product present and 100 represents the quantity of chemical present initially. For what value of x is the rate of the reaction a maximum?

Problem 91A Velocity and Acceleration When an object is dropped straight down, the distance (in feet) that it travels in t seconds is given by . Find the velocity at each of the following times. a. After 3 seconds ________________ b. After 5 seconds ________________ c. After 8 seconds ________________ d. Find the acceleration. (The answer here is a constant—the acceleration due to the influence of gravity alone near the surface of Earth.)

Problem 92A Baseball Roger Clemens, ace pitcher for many major league teams, including the Boston Red Sox, is standing on top of the 37-ft-high “Green Monster” left-field wall in Boston’s Fenway Park, to which he has returned for a visit. We have asked him to fire his famous 95 mph (140 ft per second) fastball straight up. The position equation, which gives the height of the ball at any time t, in seconds, is given by s(t) = ?16t2 + 140t + 37. Find the following. Source: Frederick Russell. a. The maximum height of the ball ________________ b. The time and velocity when the ball hits the ground

Problem 93A Height of a Ball If a cannonball is shot directly upward with a velocity of 256 ft per second, its height above the ground after t seconds is given by s(t) = 256t ? 16t2. Find the velocity and the acceleration after t seconds. What is the maximum height the cannonball reaches? When does it hit the ground?

Problem 94A Velocity and Acceleration of a Car A car rolls down a hill. Its distance (in feet) from its starting point is given by s(t) = 1.5t2 + 4t, where t is in seconds. a. How far will the car move in 10 seconds? ________________ b. What is the velocity at 5 seconds? At 10 seconds? ________________ c. How can you tell from v(t) that the car will not stop? ________________ d. What is the acceleration at 5 seconds? At 10 seconds? ________________ e. What is happening to the velocity and the acceleration as t increases?

Problem 3E Find f?(x) for the function. Then find f?(0) and f?(2). f(x) = 4x4 ? 3x3 ? 2x2 + 6