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mathematics
calculus with applications

Calculus With Applications 12th Edition Margaret L. Lial - Solutions

Determine whether each of the following statements is true or false, and explain why.The absolute maximum of a function always occurs where the derivative has a critical number.
Determine whether each statement is true or false, and explain why.If ƒ(c) is an absolute minimum, then ƒ(c) ≤ ƒ(x) for all x in the domain.
Determine whether each statement is true or false, and explain why.The equation xy2 + 3y - ln y = exy is written explicitly in terms of y.
Determine whether each statement is true or false, and explain why.If Δx is small, then dy ≈ Δy.
In a related rates problem, there can be more than two quantities that vary with time.Determine whether each of the following statements is true or false, and explain why.
Find the locations of any absolute extrema for the functions with graphs as follows. h(x) X₁ | 10 Χ2 X
Find dy/dx by implicit differentiation for the following.7x2 = 5y2 + 4xy + 1
For Exercises, find dy for the given values of x and Δx.y = 2x3 + x2 - 4x; x = 2, Δx = -0.2
Choose the correct answer. The economic order quantity formula assumes that(a) Purchase costs per unit differ due to quantity discounts.(b) Costs of placing an order vary with quantity ordered.(c) Periodic demand for the goods is known.(d) Erratic usage rates are cushioned by safety stocks.
In Exercises, use the steps shown to find nonnegative numbers x and y that satisfy the given requirements. Give the optimum value of the indicated expression.x + y = 90 and x2y is maximized.(a) Solve x + y = 180 for y.(b) Substitute the result from part (a) into P = xy, the equation for the
By sketching a graph of the function or by investigating values of the function near 0, find limxx→∞ The calculator shows no y-value when x = 0 because 0 is not in the domain of this function.However, we see from the graph that 0 = In |x|.
In Exercises, sketch the graph of a single function that has all of the properties listed.(a) Continuous everywhere except at x = -4, where there is a vertical asymptote(b) A y-intercept at y = -2(c) x-intercepts at x = -3, 1, and 4(d) f'(x) < 0 on (-∞, −5), (-4, −1), and (2, ∞)(e) f'(x)
In Exercises, sketch the graph of a single function that has all of the properties listed.(a) Continuous and differentiable everywhere except at x = -3, where it has a vertical asymptote(b) A horizontal asymptote at y = 1(c) An x-intercept at x = -2(d) A y-intercept at y = 4(e) f'(x) > 0 on the
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. a(t) 1 0 2 4 8 t
Researchers have found that the probability P that a plant will grow to radius R can be described by the equationwhere D is the density of the plants in an area. A graph in their publication shows an inflection point around R = 0.022. Find an expression for the value of R in terms of D at the
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?
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
When an object is dropped straight down, the distance (in feet) that it travels in t seconds is given by s(t) = -16t2. 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.
In a recent year, the rate of violent crimes in New York City continued to decrease, but at a slower rate than in previous years. Letting ƒ(t) be the rate of violent crime as a function of time, what does this tell you about ƒ(t), ƒ′(t), and ƒ″(t)?
The percent of concentration of a drug in the bloodstream t 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. K(t) = 4t 31² + 27
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 N(t) = 71.8e-8.96e-0.0685t. Find the inflection point and describe its importance to this research.
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
Researchers used a version of the Gompertz curve to model the growth of breast cancer tumors with the equation N(t) = ec(1-e-kt), 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.
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 L(t) = Be-ce-kt, where L(t) gives the length (in centimeters) after t years, B = 14.3032, c = 7.267963, and k = 0.670840. Find the inflection
The population of whooping cranes after t years is given by G(t) = 787/(1 + 60.8e -0.0473t).
The weight of cactus wrens in grams after t days is given by G(t) = 31.4/(1 + 12.5e -0.393t).
The percent of concentration of a certain drug in the bloodstream t hours after the drug is administered is given byFor example, after 1 hour the concentration is given by(a) Find the time at which concentration is a maximum.(b) Find the maximum concentration. K(t) = 3t 1² + 4
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
When a hardy new species is introduced into an area, the population often increases as shown in the graph in the next column. Explain the significance of the following function values on the graph.(a) f0 (b) ƒ(a) (c) fM f(t) fM fo a (a, f(a))
A study showed that the ratio of the probability that consumers choose brand A over brand B can be given bywhere a is the number of features of brand A, b is the number of features of brand B, and c is the number of common features. Show the following.(a) If a > b > c, then ƒ is increasing
A study on optimizing revenue from a website considered dividing customers into two groups based on a value x between 0 and 1, where x measures the proportion of the total bandwidth requested by a customer. Customers with a request less than x were considered low revenue, and those above x high
In economics, an index of absolute risk aversion is defined aswhere 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 U(M) = √M and for U(M) = M2/3, and determine which
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
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.
In Exercises, find the point of diminishing returns (x, y) for the given functions, where R(x), represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars).R(x) = -0.6x3 + 3.7x2 + 5x, 0 ≤ x ≤ 6
In Exercises, find the point of diminishing returns (x, y) for the given functions, where R(x), represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars).R(x) = -0.3x3 + x2 + 11.4x, 0 ≤ x ≤ 6
In Exercises, find the point of diminishing returns (x, y) for the given functions, where R(x), represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars).R(x) = 4/27 (-x3 + 66x2 + 1050x - 400), 0 ≤ x ≤ 25
Sometimes the derivative of a function is known, but not the function. We will see more of this later. For each function f defined in Exercises, 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)
In Exercises, find the point of diminishing returns (x, y) for the given functions, where R(x), represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars).R(x) = 10,000 - x3 + 42x2 + 800x, 0 ≤ x ≤ 20
Suppose a friend makes the following argument. A function ƒ is increasing and concave downward. Therefore, ƒ′ is positive and decreasing, so it eventually becomes 0 and then negative, at which point ƒ decreases. Show that your friend is wrong by giving an example of a function that is always
As we saw in Chapter 3, a group of MIT professors created a function for transforming student test scores of 0 to 100 into scores from 0 to 5, with transformed scores concentrated around 3, 4, and 5. Transformed scores from 4.5 to 5 became grades of A, 3.5 to 4.5 became B, and 2.5 to 3.5 became C.
Sometimes the derivative of a function is known, but not the function. We will see more of this later. For each function f defined in Exercises, 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)
Sometimes the derivative of a function is known, but not the function. We will see more of this later. For each function f defined in Exercises, 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)
The graph shows the total inventory of nuclear weapons held by the United States and by the Soviet Union and its successor states from 1945 to 2010.(a) In what years was the U.S. total inventory of weapons at a relative maximum?(b) When the U.S. total inventory of weapons was at the largest
Sometimes the derivative of a function is known, but not the function. We will see more of this later. For each function f defined in Exercises, 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)
A projectile is shot straight up with an initial velocity of 512 ft per second. Its height above the ground after t seconds is given by s(t) = 512t - 16t2.(a) Find the velocity and acceleration after t seconds.(b) What is the maximum height attained?(c) When does the projectile hit the ground and
Find any critical numbers for f in Exercises 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
Find any critical numbers for f in Exercises 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
Researchers used a version of the Gompertz curve to model the rate that children learn with the equation y(t) = Act, where y(t) is the portion of children of age t years passing a certain mental test, A = 0.3982 * 10-291, and c = 0.4252. Find the inflection point and describe what it signifies.
A formula proposed by Hurley for the red cell volume (RCV) in milliliters for males is RCV = 1486S2 - 4106S + 4514, where S is the surface area (in square meters). A formula given by Pearson et al. is RCV = 1486S - 825.(a) For the value of S for which the RCV values given by the two formulas are
Find any critical numbers for f in Exercises 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
The number of imagoes (sexually mature adult fruit flies) per mated female per day (y) can be approximated by y = 34.7(1.0186)-xx-0.658, where x is the mean density of the mated population (measured as flies per bottle) over a 16-day period. Sketch the graph of the function.
Find any critical numbers for f in Exercises 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
The graph below shows the horsepower and torque as a function of the engine speed for a 1964 Ford Mustang.(a) On what intervals is the power increasing with engine speed?(b) On what intervals is the power decreasing with engine speed?(c) On what intervals is the torque increasing with engine
The association between velocity during exercise and blood lactate concentration after submaximal 800-m exercise of thoroughbred racehorses on sand and grass tracks has been studied. The lactate-velocity relationship can be described by the functions where ∫1 (v) and 12 (v) are the lactate
In the FitzHugh-Nagumo model of how neurons communicate, the rate of change of the electric potential v with respect to time is given as a function of v by ƒ(v) = v(a - v)(v - 1), where a is a positive constant. Sketch a graph of this function when a = 0.25 and 0 ≤ v ≤ 1.
Find any critical numbers for f in Exercises 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
Many biological variables depend on body mass, with a functional relationship of the form Y = Y0 Mb, where M represents body mass, b is a multiple of 1/4, and Y0 is a constant. For example, when Y represents metabolic rate, b = 3/4. When Y represents heartbeat, b = -1/4. When Y represents life
Find any critical numbers for f in Exercises 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
Find any critical numbers for f in Exercises 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
An abstract for an article states, “We tentatively conclude that Olympic weightlifting ability in trained subjects undergoes a nonlinear decline with age, in which the second derivative of the performance versus age curve repeatedly changes sign.”(a) What does this quote tell you about the
Find any critical numbers for f in Exercises 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
Mathematician Hugo Rossi wrote: “In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection.” Explain what President Nixon’s announcement had to
What is true about the slope of the tangent line to the graph of ƒ(x) = ln x as x→ ∞? As x→ 0?
A survey of 185 public universities found that “the salaries and benefits of their presidents continued to rise, though at a slower rate than in years past.” Let ƒ(t) represent the total salary and compensation of the average public university president in year t. What does this statement tell
Describe the slope of the tangent line to the graph of ƒ(x) = ex for the following.(a) x→ -∞(b) x→ 0
The average price (in cents) per gallon of unleaded gasoline in the U.S. can be approximated by the function p(t) = 1.8841³ 1.8841³ 82.56t² + 1168t - - 5013, for 10 < t ≤ 19, where t is the number of years since 2000.(a) Determine the interval(s) on which the price is increasing.(b)
(a) Graph the two functions ƒ(x) = x7/3 and g(x) = x5/3 on the window [-2, 2] by [-2, 2].(b) Verify that both ƒ and g have an inflection point at (0, 0).(c) How is the value of ƒ(x) different from g(0)?(d) Based on what you have seen so far in this exercise, is it always possible to tell the
The cost function to produce q electric cat brushes is given by C(q) = -10q² + 250q. The demand equation is given by p = -q²- 3q + 299, where p is the price in dollars.(a) Find and simplify the profit function.(b) Find the number of brushes that will produce the maximum profit.(c) Find the
For each of the exercises 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 38In Exercises, find the open intervals
Give an example of a function ƒ(x) such that ƒ′(0) = 0 but ƒ″(02) does not exist. Is there a relative minimum or maximum or an inflection point at x = 0?
For each of the exercises 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 37In Exercises, find the open intervals
In Exercises, P(t) is the price of a certain stock at time t during a particular day.(a) When the stock reaches its highest price of the day, are P′(t) and P″(t) positive, zero, or negative?(b) Explain your answer.
For each of the exercises 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 36In Exercises, find the open intervals
For each of the exercises 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 35In Exercises, find the open intervals
In Exercises, P(t) is the price of a certain stock at time t during a particular day.(a) If the price of the stock is falling faster and faster, are P′(t) and P″(t) positive or negative?(b) Explain your answer.
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.ƒ(x) = 5-x2
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.ƒ(x) = x2log |x|
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.ƒ(x) = x2 + 8 ln|x + 1|
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.ƒ(x) = ln(x2 + 1)
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.ƒ(x) = x7/3 + 56x4/3
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.ƒ(x) = x8/3 - 4x5/3
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.ƒ(x) = 2e-x2
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.ƒ(x) = 18x - 18e-x
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.ƒ(x) = -x(x - 3)2
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) -2 x + 1
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points.ƒ(x) = x(x + 5)2
In Exercises, find the open intervals where the functions are concave upward or concave downward. Find any inflection points. f(x) = 3 x-5
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,
In Exercises, complete the following for each function.(a) Find intervals where the function is increasing or decreasing, and determine any relative extrema.(b) Find intervals where the function is concave upward or concave downward, and determine any inflection points.(c) Graph each function,

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