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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
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. f(x) In x x² [1,4]
In Exercises, find the equation of the tangent line at the given value of x on each curve.y3 + xy - y = 8x4; x = 1
A company determines the demand, in hundreds, for their novelty t-shirts to be D(q) = 400 - p2, where p is the price for one shirt.(a) Find the elasticity of demand E.(b) If the price is $5 per shirt, calculate the elasticity. Is the demand elastic or inelastic? Will an increase in price
In Exercises, find the equation of the tangent line at the given point on each curve. y In(x² + y²) = In 5x + X 2; (1, 2)
Your company needs to design cylindrical metal containers with a volume of 16 cubic feet. The top and bottom will be made of a sturdy material that costs $2 per square foot, while the material for the sides costs $1 per square foot. Find the radius, height, and cost of the least expensive container.
A spherical snowball is placed in the sun. The sun melts the snowball so that its radius decreases 1/4 in. per hour. Find the rate of change of the volume with respect to time at the instant the radius is 4 in.
Find an equation of the line tangent to the graph of 8y3 - 4xy2 = 20 at the point (-3, 1).
The population of bacteria (in millions) in a certain culture t hours after an experimental nutrient is introduced into the culture isUse the differential to approximate the changes in population for the following changes in t.(a) 2 to 2.5 (b) 3 to 3.25 P(t) = 25t 8 + 1². +2
The radius of a blood vessel is 1.7 mm. A drug causes the radius to change to 1.6 mm. Find the approximate change in the area of a cross section of the vessel.
Suppose the demand for tickets to an amusement park can be estimated by the function D(p) = -25p2 + 63,075, where p is the price for one ticket.(a) Find the elasticity of demand E.(b) If the price is $25 per ticket, calculate the elasticity. Is the demand elastic or inelastic? Will an increase in
For the can problem in Example 4, the minimum surface area required that the height be twice the radius. Show that this is true for a can of arbitrary volume V.Example 4A coffee company wants to manufacture cylindrical aluminum coffee cans with a volume of 1000 cm3 (1 liter). What should the radius
A rock is thrown into a still pond. The circular ripples move outward from the point of impact of the rock so that the radius of the circle formed by a ripple increases at the rate of 2 ft per minute. Find the rate at which the area is changing at the instant the radius is 4 ft.
Find the equation of the line tangent to the graph of √2y - 4xy = -22 at the point (3, 2).
A mathematics book is to contain 36 in2 of printed matter per page, with margins of 1 in. along the sides and 1(1/2) in. along the top and bottom. Find the dimensions of the page that will require the minimum amount of paper. (See the figure.) 1¹/2" 1¹/2" | 1" | I 36 in.² 1 | 1" |
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) = x2 - 8 ln x; [1, 4]
In Exercises, find the equation of the tangent line at the given point on each curve.ln1x + y2 = x3y2 + ln(x2 + 2) - 4; (1, 2)
Find the elasticity of demand (E) for the given demand function at the indicated values of p. Is the demand elastic, inelastic, or neither at the indicated values? Interpret your results.q = 300 - 2p(a) p = +100 (b) p = +50
The concentration of a certain drug in the bloodstream t hours after being administered is approximatelyUse the differential to approximate the changes in concentration for the following changes in t.(a) 1 to 1.5 (b) 2 to 2.25 C(t)= 5t 9 + 1².
A car leaves a given point and travels north at 30 mph.(a) Another car leaves the same point at the same time and travels west at 40 mph. At what rate is the distance between the two cars changing at the instant when the cars have traveled 2 hours?(b) Suppose that, in part (a), the second car left
In Exercises, find dy/dx.ln(x + y) = 1 + x2 + y3
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 + 3x2/3; [-10, 1]
In Exercises, find the equation of the tangent line at the given point on each curve.2xexy = ex3 + yex2; (1, 1)
Find the elasticity of demand (E) for the given demand function at the indicated values of p. Is the demand elastic, inelastic, or neither at the indicated values? Interpret your results.q = 400 - 0.2p2(a) p = +20 (b) p = +40
A closed box with a square base is to have a volume of 16,000 cm3. The material for the top and bottom of the box costs $3 per square centimeter, while the material for the sides costs $1.50 per square centimeter. Find the dimensions of the box that will lead to the minimum total cost. What is the
A 17-ft ladder is placed against a building. The base of the ladder is slipping away from the building at a rate of 9 ft per minute. Find the rate at which the top of the ladder is sliding down the building at the instant when the bottom of the ladder is 8 ft from the base of the building.
In Exercises, find dy/dx.ln(xy + 1) = 2xy3 + 4
Under certain conditions, a person can memorize W words in t minutes, whereFind the rate of change in the number of words memorized when t = 5. W(t) = -0.021² + t t + 1
In Exercises, find dy/dx. x + 2y X x - 3у = = ,1/2
In Exercise 61 in the section on Polynomial and Rational Functions, we gave the function defined by A(x) = 0.003631x3 - 0.03746x2 + 0.1012x + 0.009 as the approximate blood alcohol concentration in a 170-lb woman x hours after drinking 2 oz of alcohol on an empty stomach, for x in the interval 30,
In Exercises, find the equation of the tangent line at the given point on each curve.ex2+y2 = xe5y - y2e5x/2; (2, 1)
For each of the following demand functions, find(a) E, and (b) Values of q (if any) at which total revenue is maximized.q = 10 - ln p
Consider the problem of cutting corners out of a rectangle and folding up the sides to make a box. (a) In the solution to Example 3, compare the area of the base of the box with the area of the walls.(b) Repeat part (a) for the solution to Exercise 25.(c) Make a conjecture about the area of the
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) = 5x2/3 + 2x5/3; [-2, 1]
In Exercises, find dy/dx. 2 Vy - 9x2/3 + y
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. f(x) = x - 1 x² + 1² ; [1,5]
A company estimates that the revenue (in dollars) from the sale of x doghouses is given by R(x) = 12,000 1n(0.01x + 1). Use the differential to approximate the change in revenue from the sale of one more doghouse when 100 doghouses are sold.
The average cost (in dollars) to manufacture x dozen marking pencils is A(x) = 0.04x3 + 0.1x2 + 0.5x + 6. Use the differential to approximate the changes in the average cost for the following changes in x.(a) 3 to 4 (b) 5 to 6
Suppose that in the inventory problem, the storage cost depends on the maximum inventory size, rather than the average. This would be more realistic if, for example, the company had to build a warehouse large enough to hold the maximum inventory, and the cost of storage was the same no matter how
What is the difference between a relative extremum and an absolute extremum?
Why do you think that the cost g does not appear in the equation for q [Equation (3)]?
Sociologists have found that crime rates are influenced by temperature. In a midwestern town of 100,000 people, the crime rate has been approximated aswhere C is the number of crimes per month and T is the average monthly temperature in degrees Fahrenheit. The average temperature for May was 76,
In Exercises, find the equation of the tangent line at the given point on each curve. y + Vx V = 3; (4,2)
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.f(x) = (x² - 16)2/3; [-5,8]
Beach balls 1 ft in diameter have a thickness of 0.03 in. How much material would be needed to make 5000 beach balls?
In Exercises, find dy/dx. 6+5x 2 - Зу = -18 5x
For each of the following demand functions, find(a) E, and (b) Values of q (if any) at which total revenue is maximized.p = 400e-0.2q
An open box will be made by cutting a square from each corner of a 3-ft by 8-ft piece of cardboard and then folding up the sides. What size square should be cut from each corner in order to produce a box of maximum volume?
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) = (x2 - 4)1/3; [-2, 3]
A cube 4 in. on an edge is given a protective coating 0.1 in. thick. About how much coating should a production manager order for 1000 such cubes?
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. f(x) = X 2 x² + 2 ; [0,4]
In Exercises, find the equation of the tangent line at the given point on each curve.2y2 - √x = 4; (16, 2)
A manufacturing firm needs to design an open-topped box with a square base. The box must hold 32 in3. Find the dimensions of the box that can be built with the minimum amount of materials. (See the figure.)
For each of the following demand functions, find(a) E, and (b) Values of q (if any) at which total revenue is maximized.q = 48,000 - 10p2
A company wishes to manufacture a box with a volume of 36 ft3 that is open on top and is twice as long as it is wide. Find the dimensions of the box produced from the minimum amount of material.
The energy cost of horizontal locomotion as a function of the body mass of a lizard is given by E = 26.5m-0.34, where m is the mass of the lizard (in kilograms) and E is the energy expenditure (in kcal/kg/km). Suppose that the mass of a 5-kg lizard is increasing at a rate of 0.05 kg per day. Find
In Exercises, find dy/dx.9√x + 4y3 = 2√y
The cost function for the company in Exercise 23 is C(x) = 150 + 75x, where x represents the demand for the product. Find the approximate change in profit for a 1-unit change in demand when demand is at a level of 100 doghouses. Use the differential.Exercise 23A company estimates that the revenue
In Exercises, find the equation of the tangent line at the given point on each curve.x2y3 = 8; (-1, 2)
For each of the following demand functions, find(a) E, and (b) Values of q (if any) at which total revenue is maximized.q = 37,500 - 5p2
The average daily metabolic rate for captive animals from weasels to elk can be expressed as a function of mass by r = 140.2m0.75, where m is the mass of the animal (in kilograms) and r is the metabolic rate (in kcal per day).(a) Suppose that the mass of a weasel is changing with respect to time at
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. f(x) 8 + x 8 - x ; [4,6]
In Exercises, find the equation of the tangent line at the given point on each curve.x2y2 = 1; (-1, 1)
For each of the following demand functions, find(a) E, and (b) Values of q (if any) at which total revenue is maximized.q = 25,000 - 50p
Decide what you would do if your assistant presented the following contract for your signature: Your firm offers to deliver 250 tables to a dealer, at $160 per table, and to reduce the price per table on the entire order by 50¢ for each additional table over 250. Find the dollar total
The energy cost of bird flight as a function of body mass is given by E = 429m-0.35, where m is the mass of the bird (in grams) and E is the energy expenditure (in calories per gram per hour). Suppose that the mass of a 10-g bird is increasing at a rate of 0.001 g per hour. Find the rate at which
In Exercises, find dy/dx.x2y3 + 4xy = 2
In Exercises, find the equation of the tangent line at the given point on each curve.x2 + y2 = 100; (8, -6)
For each of the following demand functions, find(a) E, and (b) Values of q (if any) at which total revenue is maximized. 50 - 9 = 50 P 4
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. 1 - f(x) = x; [0,3] 3 + x -
The brain mass of a fetus can be estimated using the total mass of the fetus by the function b = 0.22m0.87, where m is the mass of the fetus (in grams) and b is the brain mass (in grams). Suppose the brain mass of a 25-g fetus is changing at a rate of 0.25 g per day. Use this to estimate the rate
In Exercises, find dy/dx.x2 - 4y2 = 3x3y4
Suppose x and y are two quantities that vary with time according to the allometric formula y = nxm. Show that the derivatives of x and y are related by the formula 1 dy y dt 1 dx x dt m-
In Exercises, find the equation of the tangent line at the given point on each curve.x2 + y2 = 25; (-3, 4)
The demand for grass seed (in thousands of pounds) at a price of p dollars is D(p) = -3p3 - 2p2 + 1500. Use the differential to approximate the changes in demand for the following changes in p.(a) $2 to $2.10 (b) $6 to $6.15
Suppose that in the inventory problem, the storage cost is a combination of the cost described in the text and the cost described in Exercise 17. In other words, suppose there is an annual cost, k1, for storing a single unit, plus an annual cost per unit, k2, that must be paid for each unit up to
Every year, Gianna sells 30,000 cases of her Famous Spaghetti Sauce. It costs her $1 per year in electricity to store a case, plus she must pay annual warehouse fees of $2 per case for the maximum number of cases she will store. If it costs her $750 to set up a production run, plus $8 per case to
In planning a restaurant, it is estimated that a profit of $8 per seat will be made if the number of seats is no more than 50, inclusive. On the other hand, the profit on each seat will decrease by 10¢ for each seat above 50.(a) Find the number of seats that will produce the maximum profit.(b)
When a term involving y is differentiated in implicit differentiation, it is multiplied by dy/dx. Why? Why aren’t terms involving x multiplied by dx/dx?
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) = x4 - 32x2 - 7; [-5, 6]
Use the differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.1n 0.98
A local club is arranging a charter flight to Hawaii. The cost of the trip is $1600 each for 90 passengers, with a refund of $10 per passenger for each passenger in excess of 90.(a) Find the number of passengers that will maximize the revenue received from the flight.(b) Find the maximum revenue.
A cross-country skier has a history of heart problems. She takes nitroglycerin to dilate blood vessels, thus avoiding angina (chest pain) due to blood vessel contraction. Use Poiseuille’s law with k = 555.6 to find the rate of change of the blood velocity when R = 0.02 mm and R is changing at
When is it necessary to use implicit differentiation?
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) = x4 - 18x2 + 1; [-4, 4]
Find dy/dx by implicit differentiation for the following.x + ln y = x2y3
Use the differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.1n 1.05
A book publisher wants to know how many times a year a print run should be scheduled. Suppose it costs $1000 to set up the printing process, and the subsequent cost per book is so low it can be ignored. Suppose further that the annual warehouse cost is $6 times the maximum number of books stored.
A fence must be built to enclose a rectangular area of 20,000 ft2. Fencing material costs $2.50 per foot for the two sides facing north and south and $3.20 per foot for the other two sides. Find the cost of the least expensive fence.
A company is increasing production at the rate of 25 units per day. The daily demand function is determined by the fact that the price (in dollars) is a linear function of q. At a price of $70, the demand is 0, and 100 items will be demanded at a price of $60. Find the rate of change of revenue
Find the absolute maximum and minimum of ƒ(x) = e2x/x2 on each interval.(a) [1/2 , 2] (b) [1, 3]
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. f(x) 1 x3 3 1 x² 2 - 6x + 3; [-4,2]
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. 3 f(x) = x² + ²x³² - 4 1 3 x² - 4x + 1; [-3,2]
Find dy/dx by implicit differentiation for the following.x2ey + y = x3
Use the differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.e-0.002
A fence must be built in a large field to enclose a rectangular area of 25,600 m2. One side of the area is bounded by an existing fence; no fence is needed there. Material for the fence costs $3 per meter for the two ends and $1.50 per meter for the side opposite the existing fence. Find the cost
The demand function for a certain product is determined by the fact that the product of the price and the quantity demanded equals 8000. The product currently sells for $3.50 per unit. Suppose manufacturing costs are increasing over time at a rate of 15% and the company plans to increase the price
Find the absolute maximum and minimum of ƒ(x) = 21nx/x2 on each interval.(a) [1, 4] (b) [2, 5]
Find dy/dx by implicit differentiation for the following.ex2y = 5x + 4y + 2
An ecologist is conducting a research project on breeding pheasants in captivity. She first must construct suitable pens. She wants a rectangular area with two additional fences across its width, as shown in the sketch. Find the maximum area she can enclose with 3600 m of fencing.
Use the differential to approximate each quantity. Then use a calculator to approximate the quantity, and give the absolute value of the difference in the two results to 4 decimal places.e0.01
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