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mathematics
calculus with applications
Calculus For Business, Economics And The Social And Life Sciences 11th Brief Edition Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price - Solutions
Rita is a postal clerk. She comes to work at 6 A.M. and t hours later has sorted approximately f (t) = −t3 + 7t2 + 200t letters. At what time during the period from 6 A.M. to 10 A.M. is Rita performing at peak efficiency?
For what value of x in the interval −1 ≤ x ≤ 4 is the graph of the functionsteepest? What is the slope of the tangent at this point? f(x) = 2x² - 13/12/²
In Exercises 1 through 4, determine where the second derivative of the function is positive and where it is negative. -ㅠ 플 y 0 2 - ㅠ 71 X
In Exercises 1 through 10, determine intervals of increase and decrease and intervals of concavity for the given function. Then sketch the graph of the function. Be sure to show all key features such as intercepts, asymptotes, high and low points, points of inflection, cusps, and vertical
In Exercises 1 through 8, determine the vertical and horizontal asymptotes of the given graph. y 3- 0 5
Determine all vertical and horizontal asymptotes for the graph of each of these functions. a. f(x) b. f(x) c. f(x) d. f(x) = 2x - 1 x + 3 X 2²-1 + x - 1 2x² + x3 1 X 1 √x
In Exercises 9 through 22, find the intervals of increase and decrease for the given function. f(x) = √6- x - x²
In Exercises 13 through 26, determine where the given function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points, and sketch the graph of the function.f(x) = (x2 − 5)3
In Exercises 17 through 22, you are given the price p(q) at which q units of a particular commodity can be sold and the total cost C(q) of producing the q units. In each case:(a) Find the revenue function R(q), the profit function P(q), the marginal revenue R'(q), and marginal cost C'(q). Sketch
In Exercises 17 through 22, you are given the price p(q) at which q units of a particular commodity can be sold and the total cost C(q) of producing the q units. In each case:(a) Find the revenue function R(q), the profit function P(q), the marginal revenue R'(q), and marginal cost C'(q). Sketch
In Exercises 9 through 22, find the intervals of increase and decrease for the given function. F(x) = x + 9 X
In Exercises 13 through 26, determine where the given function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points, and sketch the graph of the function.f(x) = x5 − 5x
In Exercises 17 through 32, sketch the graph of the given function.f(x) = x4 + 4x3 + 4x2
In Exercises 9 through 22, find the intervals of increase and decrease for the given function. f(t) t (t + 3)²
In Exercises 17 through 32, sketch the graph of the given function.f(x) = (2x − 1)2(x2 − 9)
A retailer has bought several cases of a certain imported wine. As the wine ages, its value initially increases, but eventually the wine will pass its prime and its value will decrease. Suppose that x years from now, the value of a case will be changing at the rate of 53 − 10x dollars per year.
In Exercises 17 through 20, sketch the graph of a function f that has all the given properties.a. f'(x) > 0 when x < −2 and when −2 < x < 3b. f'(x) < 0 when x > 3c. f'(−2) = 0 and f'(3) = 0
In Exercises 17 through 22, you are given the price p(q) at which q units of a particular commodity can be sold and the total cost C(q) of producing the q units. In each case:(a) Find the revenue function R(q), the profit function P(q), the marginal revenue R'(q), and marginal cost C'(q). Sketch
Each machine at a certain factory can produce 50 units per hour. The setup cost is $80 per machine, and the operating cost is $5 per hour. How many machines should be used to produce 8,000 units at the least possible cost?
In Exercises 17 through 32, sketch the graph of the given function.f(x) = 3x4 − 4x2 + 3
In Exercises 13 through 26, determine where the given function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points, and sketch the graph of the function.f(x) = (x − 2)4
In Exercises 17 through 20, sketch the graph of a function f that has all the given properties.a. f'(x) > 0 when 1 < x < 2b. f'(x) < 0 when x < 1 and when x > 2c. f"(x) > 0 for x < 2 and for x > 2d. f'(1) = 0 and f'(2) is undefined.
In Exercises 17 through 22, you are given the price p(q) at which q units of a particular commodity can be sold and the total cost C(q) of producing the q units. In each case:(a) Find the revenue function R(q), the profit function P(q), the marginal revenue R'(q), and marginal cost C'(q). Sketch
In Exercises 13 through 26, determine where the given function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points, and sketch the graph of the function.f(s) = 2s(s + 4)3
In Exercises 9 through 22, find the intervals of increase and decrease for the given function. f(x) = √x + 1 √x
In Exercises 9 through 22, find the intervals of increase and decrease for the given function. G(x) = x² - 12/1/2
In Exercises 17 through 20, sketch the graph of a function f that has all the given properties.a. f'(x) > 0 when x < 1b. f'(x) < 0 when x > 1c. f"(x) > 0 when x < 1 and when x > 1d. f'(1) is undefined.
In Exercises 17 through 32, sketch the graph of the given function. f(x) = 1 2x + 3
In Exercises 13 through 26, determine where the given function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points, and sketch the graph of the function. g(x)=√x² + 1
In Exercises 13 through 26, determine where the given function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points, and sketch the graph of the function.f(x) = (x2 − 3)3
It is estimated that the cost of constructing an office building that is n floors high is C(n) = 2n2 + 500n + 600 thousand dollars. How many floors should the building have to minimize the average cost per floor?
In Exercises 17 through 32, sketch the graph of the given function.f(x) = x3 − 3x4
In Exercises 21 through 24, find all critical numbers for the given function f(x) and use the second derivative test to determine which (if any) critical points are relative maxima or relative minima.f(x) = −2x3 + 3x2 + 12x − 5
In Exercises 21 through 24, find all critical numbers for the given function f(x) and use the second derivative test to determine which (if any) critical points are relative maxima or relative minima.f(x) = x(2x − 3)2
A manufacturer of medical monitoring devices uses 36,000 cases of components per year. The ordering cost is $54 per shipment, and the annual cost of storage is $1.20 per case. The components are used at a constant rate throughout the year, and each shipment arrives just as the preceding shipment is
In Exercises 17 through 32, sketch the graph of the given function. f(x) x + 3 x-5
In Exercises 21 through 24, find all critical numbers for the given function f(x) and use the second derivative test to determine which (if any) critical points are relative maxima or relative minima. f(x) x + 1
In Exercises 13 through 26, determine where the given function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points, and sketch the graph of the function. f(x) = 2 x² + 3
In Exercises 13 through 26, determine where the given function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points, and sketch the graph of the function. f(x) = 1 2 x + x + 1
A monopolist is a manufacturer who can manipulate the price of a commodity and usually does so with an eye toward maximizing profit. When the government taxes output, the tax effectively becomes an additional cost item, and the monopolist is forced to decide how much of the tax to absorb and how
In Exercises 21 through 24, find all critical numbers for the given function f(x) and use the second derivative test to determine which (if any) critical points are relative maxima or relative minima. f(x) = 1 X 1 x + 3
In Exercises 23 through 28, compute the elasticity of demand for the given demand function D(p) and determine whether the demand is elastic, inelastic, or of unit elasticity at the indicated price p.D(p) = −1.3p + 10; p = 4
In Exercises 17 through 32, sketch the graph of the given function. f(x) || 1² x + 2
A plastics firm has received an order from the city recreation department to manufacture 8,000 special Styrofoam kickboards for its summer swimming program. The firm owns 10 machines, each of which can produce 30 kickboards an hour. The cost of setting up the machines to produce the kickboards is
In Exercises 23 through 28, compute the elasticity of demand for the given demand function D(p) and determine whether the demand is elastic, inelastic, or of unit elasticity at the indicated price p.D(p) = − 1.5p + 25; p = 12
Paula Perkins, the owner of Paula’s Perfume Shoppe, expects to sell 800 bottles of a certain brand of perfume this year. The perfume costs $20 per bottle, the ordering fee is $10 per shipment, and the cost of storing the perfume is 40 cents per bottle per year. The perfume is sold at a constant
In Exercises 23 through 28, compute the elasticity of demand for the given demand function D(p) and determine whether the demand is elastic, inelastic, or of unit elasticity at the indicated price p. D(p) = V400 -0.01p²; p = 120
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither.f(x) = 3x4 − 8x3 + 6x2 + 2
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither.f(x) = 324x − 72x2 + 4x3
In Exercises 25 through 28, find the absolute maximum and the absolute minimum values (if any) of the given function on the specified interval. g(s) = ܐܨ 1 s+1 2 1>ܙ>
In Exercises 17 through 32, sketch the graph of the given function. f(x) = 1 x² - 9 X
In Exercises 17 through 32, sketch the graph of the given function. f(x) = 1 V1-x²
In Exercises 23 through 28, compute the elasticity of demand for the given demand function D(p) and determine whether the demand is elastic, inelastic, or of unit elasticity at the indicated price p.D(p) = 200 − p2; p = 10
In Exercises 23 through 28, compute the elasticity of demand for the given demand function D(p) and determine whether the demand is elastic, inelastic, or of unit elasticity at the indicated price p. D(p) = 3,000 P - 100; p = 10
In Exercises 13 through 26, determine where the given function is increasing and decreasing, and where its graph is concave up and concave down. Find the relative extrema and inflection points, and sketch the graph of the function.f(x) = x4 + 6x3 − 24x2 + 24
In Exercises 17 through 32, sketch the graph of the given function. f(x) = x-9 x² + 1
For speeds between 40 and 65 miles per hour, a truck gets 480/x miles per gallon when driven at a constant speed of x miles per hour. Diesel gasoline costs $3.90 per gallon, and the driver is paid $19.50 per hour. What is the most economical constant speed between 40 and 65 miles per hour at which
In Exercises 25 through 28, find the absolute maximum and the absolute minimum values (if any) of the given function on the specified interval.f(x) = −2x3 + 3x2 + 12x − 5; −3 ≤ x ≤ 3
In Exercises 17 through 32, sketch the graph of the given function. f(x) = 1 Vx X 1 -
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither.f(t) = 2t3 + 6t2 + 6t + 5
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither.f(t) = 10t6 + 24t5 + 15t4 + 3
In Exercises 23 through 28, compute the elasticity of demand for the given demand function D(p) and determine whether the demand is elastic, inelastic, or of unit elasticity at the indicated price p. D(p) || 2,000 2 P P = 5
In Exercises 25 through 28, find the absolute maximum and the absolute minimum values (if any) of the given function on the specified interval.f(t) = −3t4 + 8t3 − 10; 0 ≤ t ≤ 3
In Exercises 25 through 28, find the absolute maximum and the absolute minimum values (if any) of the given function on the specified interval. 8 f(x) = 2x + + 2; x > 0
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. f(t) t 2 + 3
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function.f(x) = x3 + 3x2 + 1
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. f(t) = t√9-t
The first derivative of a certain function is f'(x) = x(x − 1)2.a. On what intervals is f increasing? Decreasing?b. On what intervals is the graph of f concave up? Concave down?c. Find the x coordinates of the relative extrema and inflection points of f.d. Sketch a possible graph of f(x).
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither.g(x) = (x − 1)5
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither.F(x) = 3 − (x + 1)3
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function.f(x) = x4 − 2x2 + 3
The output Q at a certain factory is a function of the number L of worker-hours of labor that are used. Use calculus to prove that when the average output per worker hour is greatest, the average output is equal to the marginal output per worker-hour. You may assume without proof that the critical
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function. 1 f(x) = x + - X
Homing pigeons will rarely fly over large bodies of water unless forced to do so, presumably because it requires more energy to maintain altitude in flight in the heavy air over cool water. Suppose a pigeon is released from a boat B floating on a lake 5 miles from a point A on the shore and 13
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. h(t) = 1² 2² +1-2
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function. 18 f(x) = 2x + 1 + - X
A manufacturer estimates that when q units of a certain commodity are produced, the profit obtained is P(q) thousand dollars, where P(q) = −2q2 + 68q − 128a. Find the average profit and the marginal profit functions.b. At what level of production q̅ is average profit equal to marginal
The first derivative of a certain function is f'(x) = x2(5 − x).a. On what intervals is f increasing? Decreasing?b. On what intervals is the graph of f concave up? Concave down?c. Find the x coordinates of the relative extrema and inflection points of f.d. Sketch a possible graph of f(x).
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function.f(x) = (x2 − 9)2
A manufacturer estimates that if x units of a particular commodity are produced, the total cost will be C(x) dollars, where C(x) = x3 − 24x2 + 350x + 338a. At what level of production will the marginal cost C'(x) be minimized?b. At what level of production will the average costbe minimized?
Two industrial plants, A and B, are located 18 miles apart, and each day, respectively, emit 80 ppm (parts per million) and 720 ppm of particulate matter. Plant A is surrounded by a restricted area of radius 1 mile, while the restricted area around plant B has a radius of 2 miles. The concentration
At what point does the tangent to the curve y = 2x3 − 3x2 + 6x have the smallest slope? What is the slope of the tangent at this point?
In Exercises 17 through 32, sketch the graph of the given function.f(x) = x3/2
A box with a rectangular base is to be constructed of material costing $2/in.2 for the sides and bottom and $3/in.2 for the top. If the box is to have volume 1,215 in.3 and the length of its base is to be twice its width what dimensions of the box will minimize its cost of construction? What is the
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. g(x) = 4- 2 X 3
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function. f(x) = x² x - 2
In Exercises 23 through 34, determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. F(x) = x² X x - 1
A manufacturer can produce sunglasses at a cost of $5 apiece and estimates that if they are sold for x dollars apiece, consumers will buy 100(20 − x) sunglasses a day. At what price should the manufacturer sell the sunglasses to maximize profit?
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function. X = ( ₁ + 1 f(x) = 2
Certain hazardous waste products have the property that as the concentration of substrate (the substance undergoing change by enzymatic action) increases, there is a toxic inhibition effect. A mathematical model for this behavior is the Haldane equation*where R is the specific growth rate of the
In Exercise 31, suppose the concentration of particulate matter arriving at a point Q from each plant decreases with the reciprocal of the square of the distance between that plant and Q. With this alteration, now where should a house be located to minimize the total concentration of particulate
In Exercises 17 through 32, sketch the graph of the given function.f(x) = x4/3
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function. h(t) || 2 1 + f²
When a particular commodity is priced at p dollars per unit, consumers demand q units, where p and q are related by the equation q2 + 3pq = 22.a. Find the elasticity of demand for this commodity.b. For a unit price of $3, is the demand elastic, inelastic, or of unit elasticity?
In Exercises 33 through 38, diagrams indicating intervals of increase or decrease and concavity are given. Sketch a possible graph for a function with these characteristics. Sign of f'(x) Sign of f'(x) ++++ ++++ 0 J + -2 + 0 + -N 2 ++++ -X
A farmer wishes to enclose a rectangular pasture with 320 feet of fence. What dimensions give the maximum area ifa. The fence is on all four sides of the pasture?b. The fence is on three sides of the pasture and the fourth side is bounded by a wall?
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function.f(x) = x2(x − 5)2
A cylindrical container with no top is to be constructed for a fixed amount of money. The cost of the material used for the bottom is 3 cents per square inch, and the cost of the material used for the curved side is 2 cents per square inch. Use calculus to derive a simple relationship between the
When an electronics store prices a certain brand of stereo at p hundred dollars per set, it is found that q sets will be sold each month, where q2 + 2p2 = 41.a. Find the elasticity of demand for the stereos.b. For a unit price of p = 4 ($400), is the demand elastic, inelastic, or of unit elasticity?
In Exercises 33 through 38, diagrams indicating intervals of increase or decrease and concavity are given. Sketch a possible graph for a function with these characteristics. Sign of f'(x) Sign of f'(x) -1 ++++ + -2 0 ++++ или ++++ + 1 + 2 ++++ X X
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