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mathematics
applied calculus
Applied Calculus 6th Edition Deborah Hughes Hallett, Patti Frazer Lock, Andrew M. Gleason, Daniel E. Flath, Sheldon P. Gordon, David O. Lomen, David Lovelock, William G. McCallum, Brad G. Osgood, Andrew Pasquale - Solutions
The population (in millions) of the United States (excluding Alaska and Hawaii) t years after 1800 is given by the function f(t) in Fig. 18(a). The graphs of f′(t) and f″(t) are shown in Figs. 18(b) and 18(c).(a) What was the population in 1925?(b) Approximately when was the population 25
Sketch the following curves. y = + 2x + 1(x > 0) 1 2x
Consider the graph of g(x) in Fig. 17.(a) If g(x) is the first derivative of f(x), describe f(x) when x = 2?(b) If g(x) is the second derivative of f(x), describe f(x) when x = 2? Y 1 Figure 17 2 y = g(x) X
Sketch the following curves. y 5 + 20 X - + 3 (x > 0)
Sketch the following curves.y = x4 - 2x2
Let f (x) be a function whose derivative isShow that the graph of f (x) has an inflection point at x = 0. f'(x) = √5x² + 1. 2
Sketch the following curves.y = x4 - 4x3
Let f (x) be a function whose derivative isNote that f′(x) is always positive. Show that the graph of f (x) has an inflection point at x = 0. f'(x) = 1 1+x²
When a mutual fund company charges a fee of 0.47% on its index funds, its assets in the fund are $41 billion. And when it charges a fee of 0.18%, its assets in the fund are $300 billion. (Source: The Boston Globe.)(a) Let x % denote the fee that the company charges the index fund and A(x) its
Draw the graph of f (x) = 1/6x3 - x2 + 3x + 3 in the window [-2, 6] by [-10, 20]. It has an inflection point when x = 2, but no relative extreme points. Enlarge the window a few times to convince yourself that there are no relative extreme points anywhere. What does this tell you about f′(x)?
A car is traveling on a straight road and s(t) is the distance traveled after t hours. Match each set of information about s(t) and its derivatives with the corresponding description of the car’s motion.InformationA. s(t) is a constant function.B. s (t) is a positive constant function.C. s (t) is
The water level in a reservoir varies during the year. Let h(t) be the depth (in feet) of the water at time t days, where t = 0 at the beginning of the year. Match each set of information about h(t) and its derivatives with the corresponding description of the reservoir’s activity.InformationA.
A long rectangular sheet of metal 30 inches wide is to be made into a gutter by turning up strips vertically along the two sides (Fig. 6). How many inches should be turned up on each side to maximize the amount of water that the gutter can carry? Fold along this line 30 in Fold along this line x 30
Jane wants to drive her tractor from point A on one side of her 5-mile-wide field to a point, B, on the opposite side of the field, as shown in Fig. 7. Jane could drive her tractor directly across the field to point C and then drive 15 miles down a road to B, or she could drive to some point, P,
Find the maximum value of the function f (x) = 2 - 6x - x2, 0 ≤ x ≤ 5, and give the value of x where this maximum occurs.
Find the minimum value of the function g(t) = t2 - 6t + 9, 1 ≤ t ≤ 6.
An open rectangular box is to be 4 feet long and have a volume of 200 cubic feet. Find the dimensions for which the amount of material needed to construct the box is as small as possible.
A closed rectangular box with a square base is to be constructed using two different types of wood. The top is made of wood costing $3 per square foot and the remainder is made of wood costing $1 per square foot. If $48 is available to spend, find the dimensions of the box of greatest volume that
A small orchard yields 25 bushels of fruit per tree when planted with 40 trees. Because of overcrowding, the yield per tree (for each tree in the orchard) is reduced by 1/2 ushel for each additional tree that is planted. How many trees should be planted to maximize the total yield of the orchard?
A publishing company sells 400,000 copies of a certain book each year. Ordering the entire amount printed at the beginning of the year ties up valuable storage space and capital. However, printing the copies in several partial runs throughout the year results in added costs for setting up each
If the demand equation for a monopolist is p = 150 - .02x and the cost function is C(x) = 10x + 300, find the value of x that maximizes the profit.
Describe the way the slope changes on the graph in Exercise 8.Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously. 1 hi
A travel agency offers a boat tour of several Caribbean islands for 3 days and 2 nights. For a group of 12 people, the cost per person is $800. For each additional person above the 12-person minimum, the cost per person is reduced by $20 for each person in the group. The maximum tour group size is
Use the given information to make a good sketch of the function f(x) near x = 3.f (3) = 1, f′(3) = 0, inflection point at x = 3, f′(x) > 0 for x > 3
Properties of various functions are described next. In each case, draw some conclusion about the graph of the function. h'(3) = 4, h"(3) = 1
Outline the procedure for solving an optimization problem.
Each of the graphs of the functions in Exercises has one relative extreme point. Plot this point and check the concavity there. Using only this information, sketch the graph. [Recall that if f (x) = ax2 + bx + c, then f(x) has a relative minimum point when a > 0 and a relative maximum point when
Shakespear’s Pizza sells 1000 large vegi pizzas per week for $18 a pizza. When the owner offers a $5 discount, the weekly sales increase to 1500.(a) Assume a linear relation between the weekly sales A(x) and the discount x. Find A(x).(b) Find the value of x that maximizes the weekly revenue.(c)
A rectangular garden of area 75 square feet is to be surrounded on three sides by a brick wall costing $10 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden that minimize the cost of materials.
In the planning of a sidewalk café, it is estimated that for 12 tables, the daily profit will be $10 per table. Because of overcrowding, for each additional table the profit per table (for every table in the café) will be reduced by $.50. How many tables should be provided to maximize the profit
Use the given information to make a good sketch of the function f(x) near x = 3. 3 ƒ(3) = 4, ƒ'(3) = − ½¾/, ƒ"(3) = −2
Describe the way the slope changes on the graph in Exercise 10.Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously.
Sketch the graphs of the following functions.f(x) = -3x3 - 6x2 - 9x - 6
Properties of various functions are described next. In each case, draw some conclusion about the graph of the function. F'(2) = -1, F"(2)
How are the cost, revenue, and profit functions related?
Each of the graphs of the functions in Exercises has one relative extreme point. Plot this point and check the concavity there. Using only this information, sketch the graph. [Recall that if f (x) = ax2 + bx + c, then f(x) has a relative minimum point when a > 0 and a relative maximum point when
Use the given information to make a good sketch of the function f(x) near x = 3. f(3) = −2, ƒ'(3) = 2, ƒ"(3) = 3
Design an open rectangular box with square ends, having volume 36 cubic inches, that minimizes the amount of material required for construction.
A closed rectangular box with a square base and a volume of 12 cubic feet is to be constructed from two different types of materials. The top is made of a metal costing $2 per square foot and the remainder of wood costing $1 per square foot. Find the dimensions of the box for which the cost of
Refer to the graph in Fig. 20.(a) At which labeled points is the function increasing?(b) At which labeled points is the graph concave up?(c) Which labeled point has the most positive slope? Figure 20 y A B D E F 'y = f(x)
Properties of various functions are described next. In each case, draw some conclusion about the graph of the function. G(10) = 2, G'(10) = 0, G"(10) > 0
Sketch the graphs of the following functions.f(x) = 1 - 3x + 3x2 - x3
Refer to the graph in Fig. 19. Fill in each box of the grid with either POS, NEG, or 0. y A y = f(x) A B Figure 19 A B C f f' f"
Refer to the graph in Fig. 20.(a) At which labeled points is the function decreasing?(b) At which labeled points is the graph concave down?(c) Which labeled point has the most negative slope (that is, negative and with the greatest magnitude)? Figure 20 y A B D E F 'y = f(x)
Each of the graphs of the functions in Exercises has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph.f(x) = x3 + 6x2 + 9x
A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 150 cubic feet. The concrete for the base costs $4 per square foot, the material for the roof costs $2 per square foot, and the material for the sides costs $2.50 per square foot. Find the dimensions
Find the dimensions of the closed rectangular box with square base and volume 8000 cubic centimeters that can be constructed with the least amount of material.
Each of the graphs of the functions in Exercises has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. ح X x - cxt = (x) f
Properties of various functions are described next. In each case, draw some conclusion about the graph of the function. ƒ(4) = −2, ƒ'(4) > 0, ƒ"(4) = −1
The monthly demand equation for an electric utility company is estimated to be p = 60 - (10-5)x, where p is measured in dollars and x is measured in thousands of kilowatt-hours. The utility has fixed costs of 7 million dollars per month and variable costs of $30 per 1000 kilowatt-hours of
Use the given information to make a good sketch of the function f(x) near x = 3.f(3) = 3, f′(3) = 1, inflection point at x = 3, f′(x) < 0 for x > 3
A supermarket is to be designed as a rectangular building with a floor area of 12,000 square feet. The front of the building will be mostly glass and will cost $70 per running foot for materials. The other three walls will be constructed of brick and cement block, at a cost of $50 per running foot.
A canvas wind shelter for the beach has a back, two square sides, and a top. Find the dimensions for which the volume will be 250 cubic feet and that requires the least possible amount of canvas.
The demand equation for a company is p = 200 - 3x, and the cost function is C(x) = 75 + 80x - x2, 0 ≤ x ≤ 40.(a) Determine the value of x and the corresponding price that maximize the profit.(b) If the government imposes a tax on the company of $4 per unit quantity produced, determine the new
Sketch the graphs of the following functions.f(x) = x4 - 6x2
Each of the graphs of the functions in Exercises has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph.f(x) = x3 - 12x
A farmer has $1500 available to build an E-shaped fence along a straight river so as to create two identical rectangular pastures. (See Fig. 13.) The materials for the side parallel to the river cost $6 per foot, and the materials for the three sections perpendicular to the river cost $5 per foot.
A certain airline requires that rectangular packages carried on an airplane by passengers be such that the sum of the three dimensions (i.e., length, width, and height), is at most 120 centimeters. Find the dimensions of the squareended rectangular package of greatest volume that meets this
Draw the graph of a function y = f(x) with the stated properties.Both the function and the slope increase as x increases.
The first and second derivatives of the function f(x) have the values given in Table 1.(a) Find the x-coordinates of all relative extreme points.(b) Find the x-coordinates of all inflection points. Table 1 Values of the First Two Derivatives of a
Each of the graphs of the functions in Exercises has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. f(x) = x³ + 9x - 2 −
An athletic field [Fig. 8] consists of a rectangular region with a semicircular region at each end. The perimeter will be used for a 440-yard track. Find the value of x for which the area of the rectangular region is as large as possible. 8 Figure 8 h I I
A savings and loan association estimates that the amount of money on deposit will be 1 million times the percentage rate of interest. For instance, a 4% interest rate will generate $4 million in deposits. If the savings and loan association can loan all the money it takes in at 10% interest, what
Sketch the graphs of the following functions.f(x) = 3x4 - 6x2 + 3
Properties of various functions are described next. In each case, draw some conclusion about the graph of the function. H(0) = 0, H'(0) = 0, H"(0) = 1
Find the dimensions of the rectangular garden of greatest area that can be fenced off (all four sides) with 300 meters of fencing.
Let P(x) be the annual profit for a certain product, where x is the amount of money spent on advertising. (See Fig. 13.)(a) Interpret P(0)(b) Describe how the marginal profit changes as the amount of money spent on advertising increases.(c) Explain the economic significance of the inflection point.
An open rectangular box is to be constructed by cutting square corners out of a 16- by 16-inch piece of cardboard and folding up the flaps. [See Fig. 9.] Find the value of x for which the volume of the box will be as large as possible. 16- x x Figure 9 8 X 8 8 X X W W W
Suppose that Fig. 20 contains the graph of y = s (t), the distance traveled by a car after t hours. Is the car going faster at t = 1 or t = 2? Figure 20 Y 0 2 t
Each of the graphs of the functions in Exercises has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. f(x) = −x³ + x² + 9x
Draw the graph of a function y = f(x) with the stated properties.The function increases and the slope decreases as x increases.
In Figs. 4(a) and 4(b), the t-axis represents time in hours.(a) When is f (t) = 1?(b) Find f (5).(c) When is f (t) changing at the rate of -.08 unit per hour?(d) How fast is f (t) changing after 8 hours? 2.4 2.0 1.6 1.2 .8 .4 ܕ 2 Figure 4 H g = (t) 6 ܪܐܕܐ t 8 10 12 14
Sketch the graphs of the following functions.f(x) = (x - 3)4
Find two positive numbers, x and y, whose sum is 100 and whose product is as large as possible.
Draw the graph of a function y = f(x) with the stated properties.The function decreases and the slope increases as x increases.
The revenue for a manufacturer is R(x) thousand dollars, where x is the number of units of goods produced (and sold) and R and R are the functions given in Figs. 14(a) and 14(b).(a) What is the revenue from producing 40 units of goods?(b) What is the marginal revenue when 17.5 units of goods are
Sketch the graphs of the following functions.f(x) = (x + 2)4 - 1
Suppose that Fig. 20 contains the graph of y = v (t), the velo city of a car after t hours. Is the car going faster at t = 1 or t = 2? Figure 20 Y 0 2 t
Each of the graphs of the functions in Exercises has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph.f(x) = 2x3 - 15x2 + 36x - 24
A closed rectangular box is to be constructed with a base that is twice as long as it is wide. If the total surface area must be 27 square feet, find the dimensions of the box that will maximize the volume.
Find two positive numbers, x and y, whose product is 100 and whose sum is as small as possible.
United States electrical energy production (in trillions of kilowatt-hours) in year t (with 1900 corresponding to t = 0) is given by f (t), where f and its derivatives are graphed in Figs. 5(a) and 5(b).(a) How much electrical energy was produced in 1950?(b) How fast was energy production rising in
Use Figure 2.21 to decide which is larger in each of the following pairs.(a) Average rate of change between x = 0 and x = 2 or between x = 2 and x = 4?(b) g(1) or g(4)?(c) g'(2) or g'(4)? g(x) I I LX L 1 2 3 4 5 6 7 Figure 2.21
Figure 2.37 shows how the pumping rate of a person’s heart changes after bleeding.(a) Find the slope of the line tangent to the graph at time 2 hours. Give units.(b) Using your answer to part (a), estimate how much the pumping rate increases during the minute beginning at time 2 hours.(c) Express
Values of x and g(x) are given in the table. For what value of x does g'(x) appear to be closest to 3? 6.2 2.7 3.2 3.7 4.2 4.7 5.2 5.7 g(x) 3.4 4.4 5.0 5.4 6.0 7.4 9.0 11.0 x
An industry is being charged by the Environmental Protection Agency (EPA) with dumping unacceptable levels of toxic pollutants in a lake. Over a period of several months, an engineering firm makes daily measurements of the rate at which pollutants are being discharged into the lake. The engineers
Are the functions in Problem continuous on the given intervals? f(x)= x-1 on 0≤x
Are the functions in Problem continuous on the given intervals? f(x) = 1 x² +1 on 0≤x≤2
Values of f(x) are in the table. Where in the interval −12 ≤ x ≤ 9 does f'(x) appear to be the greatest? Least? X -12 -9 -6 -3 0 3 1.05 1.12 1.14 1.15 1.14 f(x) 1.02 6 1.12 9 1.06
A recent study reports that men who retired late developed Alzheimer’s at a later stage than those who stopped work earlier. Each additional year of employment was associated with about a six-week later age of onset. Express these results as a statement about the derivative of a function. State
The function in Figure 2.18 has f(4) = 25 and f'(4) = 1.5. Find the coordinates of the points A, B, C. C 1 3.9 A 4 B 4.2 Figure 2.18 Tangent line f(x) X
The thickness, P, in mm, of pelican eggshells depends on the concentration, c, of PCBs in the eggshell, measured in ppm (parts per million); that is, P = f(c).(a) The derivative f'(c) is negative. What does this tell you?(b) Give units and interpret f(200) = 0.28 and f'(200) = −0.0005 in terms of
Use Figure 2.19 to fill in the blanks in the following statements about the function f at point A.(a) f(_ ) = ___ (b) f'(_ ) = ___ (7,3) (7.2.3.8) Figure 2.19 Tangent line f(x)
“Winning the war on poverty” has been described cynically as slowing the rate at which people are slipping below the poverty line. Assuming that this is happening:(a) Graph the total number of people in poverty against time.(b) If N is the number of people below the poverty line at time t, what
A vehicle moving along a straight road has distance f(t) from its starting point at time t. Which of the graphs in Figure 2.31 could be f'(t) for the following scenarios? (Assume the scales on the vertical axes are all the same.)(a) A bus on a popular route, with no traffic(b) A car with no traffic
Suppose that f(t) is a function with f(25) = 3.6 and f'(25) = −0.2. Estimate f(26) and f(30).
Which of the functions described in Problems are continuous?The number of people in a village as a function of time.
A headline in the New York Times on December 14, 2014, read:“A Steep Slide in Law School Enrollment Accelerates”(a) What function is the author talking about?(b) Draw a possible graph for the function.(c) In terms of derivatives, what is the headline saying?
For a function f(x), we know that f(20) = 68 and f'(20) = −3. Estimate f(21), f(19) and f(25).
Draw a possible graph of a continuous function y = f(x) that satisfies the following three conditions:• f'(x) > 0 for 1 < x < 3• f'(x) < 0 for x < 1 and x > 3• f'(x) = 0 at x = 1 and x = 3
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