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mathematics
calculus with applications

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

Suppose x2 + y2 + 1 = 0. Use implicit differentiation to find dy/dx. Then explain why the result you got is meaningless.
Homing pigeons avoid flying over large bodies of water, preferring to fly around them instead. (One possible explanation is the fact that extra energy is required to fly over water because air pressure drops over water in the daytime.) Assume that a pigeon released from a boat 1 mile from the shore
The number of bank burglaries (entry into or theft from a bank during nonbusiness hours) in the United the closed interval [2010, 2018].(a) Give all relative maxima and minima and when they occur on the interval.(b) Give the absolute maxima and minima and when they occur on the interval.
Fruit juice will be packaged in cylindrical cans with a volume of 40 in3 each. The top and bottom of the can cost 4¢ per in2, while the sides cost 3¢ per in2. Find the radius and height of the can of minimum cost.
A company has found that its weekly profit from the sale of x units of an auto part is given by P(x) = -0.02x3 + 600x - 20,000. Production bottlenecks limit the number of units that can be made per week to no more than 150, while a long-term contract requires that at least 50 units be made each
Let eu2-v - v = 1. Find each derivative.(a) du/dv(b) dv/du
A company plans to package its product in a cylinder that is open at one end. The cylinder is to have a volume of 27π in3. What radius should the circular bottom of the cylinder have to minimize the cost of the material?
The number of bank robberies in the United States for the years 2010–2018 is given in the following figure. Consider the closed interval [2010, 2018](a) Give all relative maxima and minima and when they occur on the interval.(b) Give the absolute maxima and minima and when they occur on the
The total profit P(x) (in thousands of dollars) from the sale of x hundred thousand automobile tires is approximated by P(x) = -x3 + 9x2 + 120x - 400, x ≥ 5. Find the number of hundred thousands of tires that must be sold to maximize profit. Find the maximum profit.
Let √u + √2v + 1 = 5. Find each derivative.(a) du/dv(b) dv/du
In Exercises, refer to Exercise 44. Find f'(S0) and solve the equation f'(S0) = 1, using a calculator to find the intersection of the graphs of f'(S0) and y = 1.Find the maximum sustainable harvest if r = 0.4 and P = 500.
The packaging department of a corporation is designing a box with a square base and no top. The volume is to be 32 m3. To reduce cost, the box is to have minimum surface area. What dimensions (height, length, and width) should the box have?
In Exercises, find d2y/dx2 by implicit differentiation.x4 + y4 = 7
In Exercises, refer to Exercise 44. Find f'(S0) and solve the equation f'(S0) = 1, using a calculator to find the intersection of the graphs of f'(S0) and y = 1.Find the maximum sustainable harvest if r = 0.1 and P = 100.
The total profit (in tens of dollars) from the sale of x hundred boxes of candy is given by P(x) = -x3 + 10x2 - 12x.(a) Find the number of boxes of candy that should be sold in order to produce maximum profit.(b) Find the maximum profit.
The population of salmon next year is given by ƒ(S) = Ser(1-S/P), where S is this year’s salmon population, P is the equilibrium population, and r is a constant that depends upon how fast the population grows. The number of salmon that can be fished next year while keeping the population the
A cone has a known height of 7.284 in. The radius of the base is measured as 1.09 in., with a possible error of ± 0.007 in. Estimate the maximum error in the volume of the cone.
In Exercises, find d2y/dx2 by implicit differentiation.4x3 + y3 = -1
In Exercise 43, implicit differentiation was used to find the relative extrema. The exercise was contrived to avoid various difficulties that could have arisen. Discuss some of the difficulties that might be encountered in such problems, and how these difficulties might be resolved.Exercise
Let f(x) = e-2x. For x > 0, let P(x) be the perimeter of therectangle with vertices (0, 0), (x, 0), (x, f(x)) and (0, f(x)).Which of the following statements is true? (a) The function P has an absolute minimum but not an absolute maximum on the interval (0, ∞).(b) The function P has an
In Exercises, find the maximum sustainable harvest when S is measured in thousands. f(s) = 25S S + 2
A worker is constructing a cubical box that must contain 125 ft3, with an error of no more than 0.3 ft3. How much error could be tolerated in the length of each side to ensure that the volume is within the tolerance?
In Exercises, find d2y/dx2 by implicit differentiation.√x + √y = 2
A lake polluted by bacteria is treated with an antibacterial chemical. After t days, the number N of bacteria per milliliter of water is approximated byfor 1 ≤ t ≤ 15.(a) When during this time will the number of bacteria be a minimum?(b) What is the minimum number of bacteria during this
Information on curves in Exercises , as well as many other curves, is available on the Famous Curves section of the MacTutor History of Mathematics Archive website at www-history.mcs .st and.ac.uk/~history.The graph of 3(x² + y²)² = 25(x² - y2), shown in the figure,is a lemniscate of Bernoulli.
Suppose x and y are related by the equation -12x + x3 + y + y2 = 4(a) Find all critical points on the curve.(b) Determine whether the critical points found in part (a) are relative maxima or relative minima by taking values of x nearby and solving for the corresponding values of y.(c) Is there an
Find the absolute maximum and minimum of ƒ(x) = 2x - 3x2/3(a) On the interval [-1, 0.5]; (b) On the interval [0.5, 2].
In Exercises, find the absolute extrema if they exist, as well as all values of x where they occur. f(x) In x x3
A sphere has a radius of 5.81 in., with a possible error of ± 0.003 in. Estimate the maximum error in the volume of the sphere. Determine the relative error of the volume.
In Exercises, find d2y/dx2 by implicit differentiation.10x2 + 3y2 = 8
In Exercises, find the maximum sustainable harvest when S is measured in thousands. See Example 5.ƒ(S) = 12S0.25Example 5Suppose the spawner-recruit function for Idaho rabbits is f(S) = 2.17√S In(S + 1), whereS is measured in thousands of rabbits. Find So and the maximum sustainable harvest,
In Exercises, evaluate dy. y 3x 7 2x + 1 - x = 2, Δ.x 0.003
In Exercises, evaluate dy.y = 8 - x2 + x3; x = -1, Δx = 0.02
In Exercises, find the absolute extrema if they exist, as well as all values of x where they occur.ƒ(x) = x1nx
The radius of a circle is measured as 4.87 in., with a possible error of ± 0.040 in. Estimate the maximum error in the area of the circle. Determine the relative error of the area.
Much has been written recently about elliptic curves because of their role in Andrew Wiles’s 1995 proof of Fermat’s Last Theorem. An elliptic curve can be written in the form y2 = x3 + ax + b, where a and b are constants, and the cubic function on the right has distinct roots. Find dy/dx for
Information on curves in Exercises , as well as many other curves, is available on the Famous Curves section of the MacTutor History of Mathematics Archive website at www-history.mcs .st and.ac.uk/~history.The graph of 2(x² + y²)² = 25xy², shown in the figure, is a double folium. Find the
Epidemiologists have found a new communicable disease running rampant in College Station, Texas. They estimate that t days after the disease is first observed in the community, the percent of the population infected by the disease is approximated by for 0 ≤ t ≤ 20.(a) After how many days
In Exercises, find the absolute extrema if they exist, as well as all values of x where they occur. f(x) = X x² + 1
A worker is cutting a square from a piece of sheet metal. The specifications call for an area that is 16 cm2 with an error of no more than 0.01 cm2. How much error could be tolerated in the length of each side to ensure that the area is within the tolerance?
The graph of x2 + y2 = 100 is a circle having center at the origin and radius 10.(a) Write the equations of the tangent lines at the points where x = 6.(b) Graph the circle and the tangent lines.
Another disease hits the chronically ill town of College Station, Texas. This time the percent of the population infected by the disease t days after it hits town is approximated by p(t) = 10te-t/8 for 0 ≤ t ≤ 40.(a) After how many days is the percent of the population infected a maximum?(b)
Information on curves in Exercises , as well as many other curves, is available on the Famous Curves section of the MacTutor History of Mathematics Archive website at www-history.mcs .st and.ac.uk/~history.The graph of y²(x² + y²) = 20x2, shown in the figure, is a kappacurve. Find the equation
Describe when linear approximations are most accurate.
In Exercises, find the absolute extrema if they exist, as well as all values of x where they occur. f(x) = x - 1 x² + 2x + 6
In Exercises, find dy/dt. y || eR 1 +1 dx dt = 3, x = 1
The edge of a square is measured as 3.45 in., with a possible error of ± 0.002 in. Estimate the maximum error in the area of the square.
Repeat Exercise 37, but make point A 7 miles from point C.Exercise 37A company wishes to run a utility cable from point A on the shore (see the figure below) to an installation at point B on the island. The island is 6 miles from the shore. Point A is 9 miles from Point C, the point on the shore
What is a differential? What is it used for?
An icicle is gradually increasing in length, while maintaining a cone shape with a length 15 times the radius. Find the approximate amount that the volume of the icicle increases when the length increases from 13 cm to 13.2 cm.
Here’s another way to identify the point of unit elasticity when the demand function is a straight line.(a) Suppose the demand function has the form q = mp + b, where m < 0 and b > 0. Find the points where the line intersects the p and q axes.(b) Find the midpoint of the two intercepts
A baseball diamond forms a square, with each side 90 feet long. (See the diagram below.) A player runs from first to second base at the speed of 20 ft per second. At what rate is the runner’s distance from home plate changing when the runner is 18 ft from second base? Third base Second base Home
A company wishes to run a utility cable from point A on the shore (see the figure below) to an installation at point B on the island. The island is 6 miles from the shore. Point A is 9 miles from Point C, the point on the shore closest to Point B. It costs $400 per mile to run the cable on land and
In Exercises, find the absolute extrema if they exist, as well as all values of x where they occur.ƒ(x) = x4 - 4x3 + 4x2 + 1
In Exercises, find dy/dt. y = xe3r, dx dt -2, x = 1
A cubical crystal is growing in size. Find the approximate change in the length of a side when the volume increases from 27 cubic mm to 27.1 cubic mm.
A geometric interpretation of elasticity is as follows. Consider the tangent line to the demand curve q = ƒ(p) at the point P0 = (P0, q0). Let the point where the tangent line inter-sects the p-axis be called A, and the point where it intersects theq-axis be called B. Let P0A and P0B be the
Information on curves in Exercises, as well as many other curves, is available on the Famous Curves section of the MacTutor History of Mathematics Archive website at www-history.mcs.st and.ac.uk/~history.The graph of x2/3 + y2/3 = 2, shown in the figure, is an astroid.Find the equation of the
In Exercises, find dy/dt. x² + 5y 2y X 2: dx dt 1, x = 2, y = 0
The price along the West Coast of the United States for Japanese spruce logs (in dollars per cubic meter) based on the demand (in thousands of cubic meters per day) has been approximated by p = 0.604q2 - 20.16q + 263.067.(a) Find the elasticity of demand.In this case, elasticity will be a function
In Exercises, find the absolute extrema if they exist, as well as all values of x where they occur. 9 f(x) = 12 - x - ² X x>0
Christine is flying her kite in a wind that is blowing it east at a rate of 50 ft per minute. She has already let out 200 ft of string, and the kite is flying 100 ft above her hand. How fast must she let out string at this moment to keep the kite flying with the same speed and altitude? 100 ft
A cylindrical box will be tied up with ribbon as shown in the figure. The longest piece of ribbon available is 130 cm long, and 10 cm of that are required for the bow. Find the radius and height of the box with the largest possible volume.
In Exercises, find the absolute extrema if they exist, as well as all values of x where they occur.ƒ(x) = -3x4 + 8x3 + 18x2 + 2
In Exercises, find the equation of the tangent line at the given value of x on each curve. y (x²64) + x²/³1/3 = 12; x = 8 18
A spherical snowball is melting. Find the approximate change in volume if the radius decreases from 3 cm to 2.8 cm.
A pulley is on the edge of a dock, 8 ft above the water level. (See the figure below.) A rope is being used to pull in a boat. The rope is attached to the boat at water level. The rope is being pulled in at the rate of 1 ft per second. Find the rate at which the boat is approaching the dock at the
In Exercises, find dy/dt. y 1 + √x dx 1 1 - √x' dt Vx -4, x = 4
In Exercises, find the absolute extrema if they exist, as well as all values of x where they occur. f(x) = 2x + x² + + 1, x > 0
A spherical balloon is being inflated. Find the approximate change in volume if the radius increases from 4 cm to 4.2 cm.
The price of beef in the United States has been found to depend on the demand (measured by per capita consumption) according to the equationFind the elasticity. Is the demand for beef elastic or inelastic? How would a 1% increase in price affect the demand? 9 342.5 0.5314* p⁰.53
A study of the demand for air travel in Australia found that the demand for discount air travel from Sydney to Melbourne (in revenue passenger kilometer per capita, the product of the number of passengers traveling on a route and the distance of the route, divided by the populations of the host
In Exercises, use a graphing calculator to determine where the derivative is equal to zero.This problem is a combination of Exercises 33 and 34. We will again minimize the cost of the can, assuming that aluminum costs 3¢ per square centimeter. In addition, there is a cost of 2¢ per cm to seal the
In Exercises, find dy/dt. y 9 - 4x 3 + 2x² dx dt = -1, x = -3
trough has a triangular cross section. The trough is 6 ft across the top, 6 ft deep, and 16 ft long. Water is being pumped into the trough at the rate of 4 ft3 per minute. Find the rate at which the height of the water is increasing at the instant that the height is 4 ft. 6ift 1 16 ft I 1 6 ft
Graph each function on the indicated domain, and use the capabilities of your calculator to find the location and value of the absolute extrema. f(x) = x³ + 2x + 5 4 x + 3x³ + 10 [-3, 0]
Researchers have observed that the mass of a female (gilt) pig can be estimated by the function M(t) = -3.5 + 197.5e-e-0.01394(t - 108.4), where t is the age of the pig (in days) and M(t) is the mass of the pig (in kilograms).(a) If a particular gilt is 80 days old, use differentials to estimate
In Exercises, find the equation of the tangent line at the given value of x on each curve. 2y³(x − 3) + x √y = 3; x = 3
Accurate methods of estimating the age of gray wolves are important to scientists who study wolf population dynamics. One method of estimating the age of a gray wolf is to measure the percent closure of the pulp cavity of a canine tooth and then estimate age bywhere p is the percent closure and
In Exercises, use a graphing calculator to determine where the derivative is equal to zero.In this modification of the can problem in Example 4, the cost must be minimized. Assume that aluminum costs 3¢ per square centimeter, and that there is an additional cost of 1¢ per cm times the height of
The Valve Corporation, a software entertainment company, ran a holiday sale on its popular Steam software program. Using data collected from the sale, it is possible to estimate the demand corresponding to various discounts in the price of the software. Assuming that the original price was $40, the
In Exercises, use a graphing calculator to determine where the derivative is equal to zero.Modify the can problem in Example 4 so the cost must be minimized. Assume that aluminum costs 3¢ per square centimeter, and that there is an additional cost of 2¢ per cm times the perimeter of the top, and
In Exercises, find dy/dt. y = 8x³ - 7x²; dx dt 4, x = 2
Graph each function on the indicated domain, and use the capabilities of your calculator to find the location and value of the absolute extrema. = -5x4 + 2x³ + 3x² + 9 2 x²x³ + x² + 7 ; [-1,1]
A streetlight is mounted at the top of a 15 ft pole. A man 6 ft tall is walking away from the streetlight at the rate of 5 ft per second. See the diagram below.(a) How fast is the length of his shadow increasing when he is 25 ft from the pole?(b) How fast is the tip of his shadow moving when he is
The shape of a colony of bacteria on a Petri dish is circular. Find the approximate increase in its area if the radius increases from 20 mm to 22 mm.
In Exercises, find the equation of the tangent line at the given value of x on each curve.y3 + xy2 + 1 = x + 2y2; x = 2
The demand for rice in Japan for a particular year was estimated by the general function q = ƒ(p) = Ap-0.13, where p represents the price of a unit of rice, A represents a constant that can be calculated uniquely for a particular year, and q represents the annual per capita rice demand. Calculate
A real estate developer has determined that lots such as those described in Exercise 13 can be sold for $0.50 per square meter. The demand is proportional to the length of the property along the river; the number he can sell in a year is numerically half of that length. What property dimensions
A sand storage tank used by the highway department for winter storms is leaking. As the sand leaks out, it forms a conical pile. The radius of the base of the pile increases at the rate of 0.75 in. per minute. The height of the pile is always twice the radius of the base. Find the rate at which the
Why is implicit differentiation used in related rates problems?
Find the absolute extrema if they exist, as well as all values of x where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.ƒ(x) = x2e-0.5x; [2, 5]
An oil slick is in the shape of a circle. Find the approximate increase in the area of the slick if its radius increases from 1.2 miles to 1.4 miles.
In Exercises, find the equation of the tangent line at the given value of x on each curve.y3 + 2x2y - 8y = x3 + 19; x = 2
The demand for crude oil in the United States was recently approximated by q = ƒ(p) = 2,431,129p-0.06, where p represents the price of crude oil in dollars per barrel and q represents the per capita consumption of crude oil. Calculate the elasticity of demand when the price is $100 per barrel. Is
A company sells square carpets for $5 per square foot. It has a simplified manufacturing process for which all the carpets each week must be the same size, and the length must be a multiple of a half foot. It has found that it can sell 100 carpets in a week when the carpets are 2 ft by 2 ft, the
An ice cube that is 3 cm on each side is melting at a rate of 2 cm3 per min. How fast is the length of the side decreasing?
What is the difference between a related rates problem and an applied extremum problem?
Find the absolute extrema if they exist, as well as all values of x where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.ƒ(x) = x + e-3x; [-1, 3]
A tumor is approximately spherical. If the radius of the tumor changes from 14 mm to 16 mm, find the approximate change in volume.

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