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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
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.S(t) = (t2 − 1)4
In Exercises 35 through 44, use calculus to sketch the graph of the given function.f(x) = x3 − 3x2
Bernardo is a real estate developer. He estimates that if 60 luxury houses are built in a certain area, the average profit will be $47,500 per house. The average profit will decrease by $500 per house for each additional house built in the area. How many houses should Bernardo build to maximize the
An art gallery offers 50 prints by a famous artist. If each print in the limited edition is priced at p dollars, it is expected that q = 500 − 2p prints will be sold.a. What limitations are there on the possible range of the price p?b. Find the elasticity of demand. Determine the values of p for
In Exercises 35 through 44, use calculus to sketch the graph of the given function.f(x) = 3x4 − 4x3
An experimental garden plot contains N annual plants, each of which produces S seeds that are dropped within the same plot. A botanical model measures the number of offspring plants A(N) that survive until the next year by the functionwhere c and p are positive constants.a. For what value of N is
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function. f(s) = s + 1 (s − 1)²
Gina manages a company that makes toys. Her firm produces an inexpensive doll (Floppsy) and an expensive doll (Moppsy) in units of x hundreds and y hundreds, respectively. Gina finds that it is possible to produce the dolls in such a way thatfor 0 ≤ x ≤ 8 and that the company receives twice as
In Exercises 27 through 38, use the second derivative test to find the relative maxima and minima of the given function. f(x) = (x - 2)³ 1²
The concentration of a drug in a patient’s bloodstream t hours after it is injected is given bymilligrams per cubic centimeter.a. Sketch the graph of the concentration function.b. At what time is the concentration decreasing most rapidly?c. What happens to the concentration in the long run (as t
A rectangle is inscribed in a right triangle, as shown in the accompanying figure. If the triangle has sides of length 5, 12, and 13, what are the dimensions of the inscribed rectangle of greatest area? 13 12 5
In Exercises 1 through 16, find the absolute maximum and absolute minimum (if any) of the given function on the specified interval. f(t) 11121₁1-2515 - 12/2 t 1'
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 tangents.
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) + 2 unu ++++ 1 X 3 ++++ X
In Exercises 1 through 8, determine the vertical and horizontal asymptotes of the given graph. -31 0 12 y = 1 Ex
In Exercises 35 through 44, use calculus to sketch the graph of the given function.f(x) = 3x4 − 8x3 + 6x2 + 2
An airline determines that when a round-trip ticket between Los Angeles and San Francisco costs p dollars (0 ≤ p ≤ 160), the daily demand for tickets is q = 256 − 0.01p2.a. Find the elasticity of demand. Determine the values of p for which the demand is elastic, inelastic, and of unit
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 tangents.
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 5 through 12, determine where the graph of the given function is concave upward and concave downward. Find the coordinates of all inflection points. g(t) = 1² 1 t
It is estimated that between the hours of noon and 7:00 P.M., the speed of highway traffic flowing past a certain downtown exit is approximately S(t) = t3 − 9t2 + 15t + 45 miles per hour, where t is the number of hours past noon. At what time between noon and 7:00 P.M. is the traffic moving
A triangle is positioned with its hypotenuse on a diameter of a circle, as shown in the accompanying figure. If the circle has radius 4, what are the dimensions of the triangle of greatest area?
In Exercises 1 through 16, find the absolute maximum and absolute minimum (if any) of the given function on the specified interval. 1 g(x) = x + - 1 x 2 ≤x≤3
In Exercises 9 through 16, find all vertical and horizontal asymptotes of the graph of the given function. f(x): = 3x - 1 x + 2
In Exercises 1 through 16, find the absolute maximum and absolute minimum (if any) of the given function on the specified interval. g(x) || 1 -9 ;0≤x≤2
A manufacturer can produce MP3 players at a cost of $90 apiece. It is estimated that if the MP3 players are sold for x dollars apiece, consumers will buy 20(180 − x) of them each month. What unit price should the manufacturer charge to maximize profit?
In Exercises 9 through 16, find all vertical and horizontal asymptotes of the graph of the given function. f(x) X 2- x
In Exercises 9 through 22, find the intervals of increase and decrease for the given function.f(x) = x2 − 4x + 5
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) = -=-x³- 3 - 9x + 2
In Exercises 9 through 16, find all vertical and horizontal asymptotes of the graph of the given function. f(x) = x² + 2 2 x²² +1
In Exercises 5 through 12, determine where the graph of the given function is concave upward and concave downward. Find the coordinates of all inflection points.F(x) = (x − 4)7/3
In Exercises 1 through 16, find the absolute maximum and absolute minimum (if any) of the given function on the specified interval. f(u) = 2u + 32 u u -; u > 0
In Exercises 11 and 12, one of the two curves shown is the graph of a certain function f(x) and the other is the graph of its derivative f'(x). Determine which curve is the graph of the derivative, and give reasons for your decision. 3 O (a) -X f X (b)
A store has been selling a popular computer game at the price of $40 per unit, and at this price, players have been buying 50 units per month. The owner of the store wishes to raise the price of the game and estimates that for each $1 increase in price, three fewer units will be sold each month. If
In Exercises 1 through 16, find the absolute maximum and absolute minimum (if any) of the given function on the specified interval. 1 f(u) = u +;u > 0 И
In Exercises 9 through 22, find the intervals of increase and decrease for the given function.f(t) = t3 + 3t2 + 1
In Exercises 9 through 16, find all vertical and horizontal asymptotes of the graph of the given function. f(t) = t² + 3t - 5 t-5t + 6
In Exercises 9 through 16, find all vertical and horizontal asymptotes of the graph of the given function. f(t) t+2 2²
In Exercises 9 through 22, find the intervals of increase and decrease for the given function. f(x) 1 3 - 9x + 2
In Exercises 11 and 12, one of the two curves shown is the graph of a certain function f(x) and the other is the graph of its derivative f'(x). Determine which curve is the graph of the derivative, and give reasons for your decision. h A 2 X 3 (b) X
In Exercises 9 through 16, find all vertical and horizontal asymptotes of the graph of the given function. g(x) || 5x² x² - 3x - 4
In Exercises 9 through 16, find all vertical and horizontal asymptotes of the graph of the given function. h(x) || 1 1 1 x-1 X X
In Exercises 13 through 16, the derivative f'(x) of a function is given. Use this information to classify each critical number of f(x) as a relative maximum, a relative minimum, or neither. f'(x) = x(x - 2)² 4 x* + 1
In Exercises 5 through 12, determine where the graph of the given function is concave upward and concave downward. Find the coordinates of all inflection points.f(x) = x4 − 6x3 + 7x − 5
In Exercises 9 through 22, find the intervals of increase and decrease for the given function.f(x) = x3 − 3x − 4
In Exercises 1 through 16, find the absolute maximum and absolute minimum (if any) of the given function on the specified interval. f(x) == ; x > 0 X
In Exercises 5 through 12, determine where the graph of the given function is concave upward and concave downward. Find the coordinates of all inflection points.g(x) = 3x5 − 25x4 + 11x − 17
In Exercises 13 through 16, the derivative f'(x) of a function is given. Use this information to classify each critical number of f(x) as a relative maximum, a relative minimum, or neither. f'(x) = x² + 2x - 3 x²(x² + 1) 2 X
In Exercises 9 through 22, find the intervals of increase and decrease for the given function. g(t) 1 t + 1 2 1 (t² + 1)²
In Exercises 9 through 22, find the intervals of increase and decrease for the given function. f(t) 1 4-2
In Exercises 1 through 16, find the absolute maximum and absolute minimum (if any) of the given function on the specified interval. f(x) = 1 it > 0
In Exercises 13 through 16, the derivative f'(x) of a function is given. Use this information to classify each critical number of f(x) as a relative maximum, a relative minimum, or neither.f(x) = x3 (2x − 3)2 (x + 1)5 (x − 7)
In Exercises 9 through 16, find all vertical and horizontal asymptotes of the graph of the given function. g(t) = t VP-4
Granville Thomas is a citrus grower in Florida. He estimates that if 60 orange trees are planted, the average yield will be 400 oranges per tree. The average yield will decrease by 4 oranges per tree for each additional tree planted on the same acreage. How many trees should Granville plant to
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) = x3 + 3x2 + 1
In Exercises 9 through 22, find the intervals of increase and decrease for the given function.g(t) = t5 − 5t4 + 100
In Exercises 1 through 16, find the absolute maximum and absolute minimum (if any) of the given function on the specified interval. f(x) = 1 0=xsl+x
In Exercises 1 through 16, find the absolute maximum and absolute minimum (if any) of the given function on the specified interval. f(x): || 1 (x + 1)² x ≥ 0
Farmers can get $8 per bushel for their potatoes on July 1, and after that, the price drops by 8 cents per bushel per day. On July 1, a farmer has 80 bushels of potatoes in the field and estimates that the crop is increasing at the rate of 1 bushel per day. When should the farmer harvest the
A baseball card store can obtain Mel Schlabotnic rookie cards at a cost of $5 per card. The store has been offering the cards at $10 apiece and, at this price, has been selling 25 cards per month. The store is planning to lower the price to stimulate sales and estimates that for each 25-cent
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 − 4x3 + 10
In Exercises 9 through 22, find the intervals of increase and decrease for the given function.f(x) = 3x5 − 5x3
In Exercises 17 through 32, sketch the graph of the given function.f(x) = x3 + 3x2 − 2
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) = x3 − 3x2 + 3x + 1
A manufacturer has been selling flashlights at $6 apiece, and at this price, consumers have been buying 3,000 flashlights per month. The manufacturer wishes to raise the price and estimates that for each $1 increase in the price, 1,000 fewer flashlights will be sold each month. The manufacturer can
A cable is to be run from a power plant on one side of a river 1,200 meters wide to a factory on the other side, 1,500 meters downstream. The cost of running the cable under the water is $25 per meter, while the cost over land is $20 per meter. What is the most economical route over which to run
In Exercises 17 through 20, sketch the graph of a function f that has all the given properties.a. f'(x) > 0 when x < 0 and when x > 5b. f'(x) < 0 when 0 < x < 5c. f"(x) > 0 when −6 < x < −3 and when x > 2d. f"(x) < 0 when x < −6 and when
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)3
In Exercises 9 through 22, find the intervals of increase and decrease for the given function. h(u) = V9 – ư
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 32, sketch the graph of the given function.f(x) = x5 − 5x4 + 93
Find the most economical route in Exercise 17 if the power plant is 2,000 meters downstream from the factory.Data from Exercises 17A cable is to be run from a power plant on one side of a river 1,200 meters wide to a factory on the other side, 1,500 meters downstream. The cost of running the cable
A cylindrical can is to hold 4π cubic inches of frozen orange juice. The cost per square inch of constructing the metal top and bottom is twice the cost per square inch of constructing the cardboard side. What are the dimensions of the least expensive can?
In physical chemistry, it is shown that the pressure P of a gas is related to the volume V and temperature T by van der Waals’ equation:where a, b, n, and R are constants. The critical temperature Tc of the gas is the highest temperature at which the gaseous and liquid phases can exist as
Dan wants to use 300 meters of fencing to surround two identical adjacent rectangular plots, as shown in the accompanying figure. How should he do this to make the combined area of the plots as large as possible? HHH 7 O
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 . - 1 ++++ X X
In Exercises 39 through 42, the derivative f'(x) of a differentiable function f (x) is given. In each case,(a) Find intervals of increase and decrease for f(x).(b) Determine values of x for which relative maxima and minima occur on the graph of f(x).(c) Find f"(x), and determine intervals of
Suppose q > 0 units of a commodity are produced at a total cost of C(q) dollars and an average cost ofIn this section, we showed that q = qc satisfies A'(qc) = 0 if and only if C'(qc) = A(qc); that is, when marginal cost equals average cost. The purpose of this problem is to show that A(q) is
In Exercises 39 through 42, the derivative f'(x) of a differentiable function f (x) is given. In each case,(a) Find intervals of increase and decrease for f(x).(b) Determine values of x for which relative maxima and minima occur on the graph of f(x).(c) Find f"(x), and determine intervals of
Use the fact that 12 fluid ounces is approximately 6.89π cubic inches to find the dimensions of the 12-ounce soda can that can be constructed using the least amount of metal. Compare these dimensions with those of one of the soda cans in your refrigerator. What do you think accounts for the
In Exercises 35 through 44, use calculus to sketch the graph of the given function.f(t) = 2t3 + 6t2 + 6t + 5
In Exercises 39 through 42, the derivative f'(x) of a differentiable function f (x) is given. In each case,(a) Find intervals of increase and decrease for f(x).(b) Determine values of x for which relative maxima and minima occur on the graph of f(x).(c) Find f"(x), and determine intervals of
Suppose that q = 500 − 2p units of a certain commodity are demanded when p dollars per unit are charged, for 0 ≤ p ≤ 250.a. Determine where the demand is elastic, inelastic, and of unit elasticity with respect to price.b. Use the results of part (a) to determine the intervals of increase and
In Exercises 39 through 42, the derivative f'(x) of a differentiable function f (x) is given. In each case,(a) Find intervals of increase and decrease for f(x).(b) Determine values of x for which relative maxima and minima occur on the graph of f(x).(c) Find f"(x), and determine intervals of
Kamal, the owner of a print shop, receives an order to produce a rectangular poster containing 648 square centimeters of print surrounded by margins of 2 centimeters on each side and 4 centimeters on the top and bottom. What dimensions should Kamal choose to minimize the total area of the poster
Suppose that the demand equation for a certain commodity is q = 60 − 0.1p (for 0 ≤ p ≤ 600).a. Express the elasticity of demand as a function of p.b. Calculate the elasticity of demand when the price is p = 200. Interpret your answer.c. At what price is the elasticity of demand equal to 1?
Find constants A and B so that the graph of the functionwill have x = 2 as a vertical asymptote and y = 4 as a horizontal asymptote. Once you find A and B, sketch the graph of f(x). f(x) = Ax - 3 5 + Bx
In Exercises 35 through 44, use calculus to sketch the graph of the given function.f(x) = x3(x + 5)2
In Exercises 35 through 44, use calculus to sketch the graph of the given function. f(x) = x + 1 x² + x + 1 2
In Exercises 39 through 42, the second derivative f"(x) of a function is given. In each case, use this information to determine where the graph of f(x) is concave upward and concave downward and find all values of x for which an inflection point occurs. [You are not required to find f(x) or the y
In Exercises 39 through 42, the second derivative f"(x) of a function is given. In each case, use this information to determine where the graph of f(x) is concave upward and concave downward and find all values of x for which an inflection point occurs. [You are not required to find f(x) or the y
Suppose that the demand equation for a certain commodity is q = 200 − 2p2 (for 0 ≤ p ≤ 10).a. Express the elasticity of demand as a function of p.b. Calculate the elasticity of demand when the price is p = 6. Interpret your answer.c. At what price is the elasticity of demand equal to 1?
Suppose the demand for a certain commodity is given by q = b − ap, where a and b are positive constants, and 0 ≤ p ≤ b/a.a. Express elasticity of demand as a function of p.b. Show that the demand is of unit elasticity at the midpoint p = b/2a of the interval 0 ≤ p ≤ b/a.c. For what values
In Exercises 39 through 42, the second derivative f"(x) of a function is given. In each case, use this information to determine where the graph of f(x) is concave upward and concave downward and find all values of x for which an inflection point occurs. f"(x) || x² + x = 2 x²2² +4
Find constants A and B so that the graph of the functionwill have x = 4 as a vertical asymptote and y = −1 as a horizontal asymptote. Once you find A and B, sketch the graph of f(x). f(x) || Ax + 2 8 Bx
In Exercises 43 through 46, the first derivative f'(x) of a certain function f(x) is given. In each case,(a) Find intervals on which f is increasing and decreasing.(b) Find intervals on which the graph of f is concave up and concave down.(c) Find the x coordinates of the relative extrema and
In Exercises 35 through 44, use calculus to sketch the graph of the given function. H(x) -(3x+- 50 = -(3x48x³ 90x² + 70) -
In Exercises 39 through 42, the second derivative f"(x) of a function is given. In each case, use this information to determine where the graph of f(x) is concave upward and concave downward and find all values of x for which an inflection point occurs. [You are not required to find f(x) or the y
A cylindrical container with no top is to be constructed to hold a fixed volume of liquid. The cost of the material used for the bottom is 3 cents per square inch, and that for the curved side is 2 cents per square inch. Use calculus to derive a simple relationship between the radius and height of
The total cost of producing x units of a particular commodity is C thousand dollars, where C(x) = 3x2 + x + 48, and the average cost isa. Find all vertical and horizontal asymptotes of the graph of A(x).b. Note that as x gets larger and larger, the term 48/x in A(x) gets smaller and smaller. What
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