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All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)

Suppose we know that the point (3, 5) is a point on the graph of a function f. Explain how to find f(3) and f1(5).8RCC 9RCC 10RCC 11RCC (a) What is an even function? How can you tell that a function is even by looking at its graph? Give an example of an even function. (b) What is an odd function? How can you tell that a function is odd by looking at its graph? Give an example of an odd function.Suppose that f has domain A and g has domain B. What are the domains of the following functions? (a) Domain of f + g (b) Domain of fg (c) Domain of f/g 14RCC 15RCC 1RE 2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 30RE 31RE 32RE 33RE 34RE 35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE Graphing Functions Determine which viewing rectangle produces the most appropriate graph of the function. 44. f(x)=100x3. (i) [4, 4] by [4, 4] (ii) [10, 10] by [10, 10] (iii) [10, 10] by [10, 40] (iv) [100, 100] by [100, 100]45RE 46RE 47RE 48RE 49RE 50RE 51RE 52RE 53RE 54RE 55RE 56RE 57RE 58RE 59RE 60RE 61RE 62RE 63RE 64RE 65RE 66RE 67RE 68RE 69RE 70RE 71RE 72RE 73RE 74RE 75RE 76RE 77RE 78RE 79RE Maximum Profit The profit P (in dollars) generated by selling x units of a certain commodity is given by P(x)=1500+12x0.0004x2 What is the maximum profit, and how many units must be sold to generate it?81RE 82RE 83RE 84RE 85RE 86RE 87RE 88RE 89RE 90RE 91RE 92RE One-to-One Functions Determine whether the function is one-to-one. 93. p(x) = 3.3 + 1.6x 2.5x3 94RE 95RE 96RE 97RE 98RE 99RE 100RE 101RE 102RE 1T 2T A function f has the following verbal description: Subtract 2, then cube the result. (a) Find a formula that expresses f algebraically. (b) Make a table of values of f, for the inputs 1, 0, 1, 2, 3, and 4. (c) Sketch a graph of f, using the table of values from part (b) to help you. (d) How do we know that f has an inverse? Give a verbal description for f1. (e) Find a formula that expresses f1 algebraically.4T A school fund-raising group sells chocolate bars to help finance a swimming pool for their physical education program. The group finds that when they set their price at x dollars per bar (where 0 x 5), their total sales revenue (in dollars) is given by the function R(x) = 500x2 + 3000x. (a) Evaluate R(2) and R(4). What do these values represent? (b) Use a graphing calculator to draw a graph of R. What does the graph tell us about what happens to revenue as the price increases from 0 to 5 dollars? (c) What is the maximum revenue, and at what price is it achieved?Determine the net change and the average rate of change for the function f(t) = t2 2t between t = 2 and t = 2 + h.7T 8T 9T 10T 11T 12T 13T 14T 15T 16T 17T 18T 19T 20T 21T 22T 1P 2P 3P 4P 5P 6P 7P 8P 9P 10P 11P Length A woman 5 ft tall is standing near a street lamp that is 12 ft tall, as shown in the figure. Find a function that models the length L of her shadow in terms of her distance d from the base of the lamp.Distance Two ships leave port at the same time. One sails south at 15 mi/h, and the other sails east at 20 mi/h. Find a function that models the distance D between the ships in terms of the time t (in hours) elapsed since their departure.14P Area An isosceles triangle has a perimeter of 8 cm. Find a function that models its area A in terms of the length of its base b.16P Area A rectangle is inscribed in a semicircle of radius 10, as shown in the figure. Find a function that models the area A of the rectangle in terms of its height h.Height The volume of a cone is 100 in3. Find a function that models the height h of the cone in terms of its radius r.Maximizing a Product Consider the following problem: Find two numbers whose sum is 19 and whose product is as large as possible. (a) Experiment with the problem by making a table like the one following, showing the product of different pairs of numbers that add up to 19. On the basis of the evidence in your table, estimate the answer to the problem. First number Second number Product 1 18 18 2 17 34 3 16 48 (b) Find a function that models the product in terms of one of the two numbers. (c) Use your model to solve the problem, and compare with your answer to part (a).Minimizing a Sum Find two positive numbers whose sum is 100 and the sum of whose squares is a minimum.Fencing a Field Consider the following problem: A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river (see the figure). What are the dimensions of the field of largest area that he can fence? (a) Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration, and use your results to estimate the dimensions of the largest possible field. (b) Find a function that models the area of the field in terms of one of its sides. (c) Use your model to solve the problem, and compare with your answer to part (a).Dividing a Pen A rancher with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle (see the figure). (a) Find a function that models the total area of the four pens. (b) Find the largest possible total area of the four pens.Fencing a Garden Plot A property owner wants to fence a garden plot adjacent to a road, as shown in the figure. The fencing next to the road must be sturdier and costs 5 per foot, but the other fencing costs just 3 per foot. The garden is to have an area of 1200 ft2. (a) Find a function that models the cost of fencing the garden. (b) Find the garden dimensions that minimize the cost of fencing. (c) If the owner has at most 600 to spend on fencing, find the range of lengths he can fence along the road.Maximizing Area A wire 10 cm long is cut into two pieces, one of length x and the other of length 10 x, as shown in the figure. Each piece is bent into the shape of a square. (a) Find a function that models the total area enclosed by the two squares. (b) Find the value of x that minimizes the total area of the two squares.Light from a Window A Norman window has the shape of a rectangle surmounted by a semicircle, as shown in the figure to the left. A Norman window with perimeter 30 ft is to be constructed. (a) Find a function that models the area of the window. (b) Find the dimensions of the window that admits the greatest amount of light.Volume of a Box A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides (see the figure). (a) Find a function that models the volume of the box. (b) Find the values of x for which the volume is greater than 200 in3. (c) Find the largest volume that such a box can have.Area of a Box An open box with a square base is to have a volume of 12 ft3. (a) Find a function that models the surface area of the box. (b) Find the box dimensions that minimize the amount of material used.Inscribed Rectangle Find the dimensions that give the largest area for the rectangle shown in the figure. Its base is on the x-axis, and its other two vertices are above the x-axis, lying on the parabola y = 8 x2.Minimizing Costs A rancher wants to build a rectangular pen with an area of 100 m2. (a) Find a function that models the length of fencing required. (b) Find the pen dimensions that require the minimum amount of fencing.Minimizing Time A man stands at a point A on the bank of a straight river, 2 mi wide. To reach point B, 7 mi downstream on the opposite bank, he first rows his boat to point P on the opposite bank and then walks the remaining distance x to B, as shown in the figure. He can row at a speed of 2 mi/h and walk at a speed of 5 mi/h. (a) Find a function that models the time needed for the trip. (b) Where should he land so that he reaches B as soon as possible?Bird Flight A bird is released from point A on an island, 5 mi from the nearest point B on a straight shoreline. The bird flies to a point C on the shoreline and then flies along the shoreline to its nesting area D (see the figure). Suppose the bird requires 10 kcal/mi of energy to fly over land and 14 kcal/mi to fly over water. (a) Use the fact that energyused=energypermilemilesflown to show that the total energy used by the bird is modeled by the function E(x)=14x2+25+10(12x) (b) If the bird instinctively chooses a path that minimizes its energy expenditure, to what point does it fly?Area of a Kite A kite frame is to be made from six pieces of wood. The four pieces that form its border have been cut to the lengths indicated in the figure. Let x be as shown in the figure (a) Show that the area of the kite is given by the function A(x)=x(25x2+144x2) (b) How long should each of the two crosspieces be to maximize the area of the kite?To put the quadratic function f(x)=ax2+bx+c in standard form, we complete the_______________.The quadratic function f(x) = a(x - h)2 + k is in standard form. (a) The graph of f is a parabola with vertex ( ______, ______ ). (b) If a 0, the graph of f opens. In this case f (h) = k is thevalue of f. (c) If a 0, the graph of f opens. In this case f (h) = k is thevalue of f.The graph of f(x) = 3(x - 2)2 - 6 is a parabola that opens , with its vertex at (,), and f (2) = is the (minimum/maximum) value of f.The graph of f(x) = -3(x - 2)2 - 6 is a parabola that opens______,with its vertex at ( ____,____ ), and f (2) = is the (minimum/maximum) value of f.Graphs of Quadratic Functions The graph of a quadratic function f is given, (a) Find the coordinates of the vertex and the x- and y-intercepts. (b) Find the maximum or minimum value of f. (c) Find the domain and range of f. 5. f(x) = x2 + 6x 5 Graphs of Quadratic Functions The graph of a quadratic function f is given, (a) Find the coordinates of the vertex and the x- and y-intercepts. (b) Find the maximum or minimum value of f. (c) Find the domain and range of f. 6. f(x)=12x22x+6 Graphs of Quadratic Functions The graph of a quadratic function f is given, (a) Find the coordinates of the vertex and the x- and y-intercepts. (b) Find the maximum or minimum value of f. (c) Find the domain and range of f. 7. f(x) = 2x2 4x 1 Graphs of Quadratic Functions The graph of a quadratic function f is given, (a) Find the coordinates of the vertex and the x- and y-intercepts. (b) Find the maximum or minimum value of f. (c) Find the domain and range of f. 8. f(x) = 3x2 + 6x 1 Graphing Quadratic Functions A quadratic function f is given. (a) Express f in standard form. (b) Find the vertex and x- and y-intercepts of f. (c) Sketch a graph of f. (d) Find the domain and range of f. 9. f(x) = x2 2x + 3 10E 11E 12E 13E Graphing Quadratic Functions A quadratic function f is given. (a) Express f in standard form. (b) Find the vertex and x- and y-intercepts of f. (c) Sketch a graph of f. (d) Find the domain and range of f. 14. f(x) = x2 + 10x 15E 16E 17E 18E Graphing Quadratic Functions A quadratic function f is given. (a) Express f in standard form. (b) Find the vertex and x- and y-intercepts of f. (c) Sketch a graph of f. (d) Find the domain and range of f. 19. f(x) = 2x2 + 4x + 3 Graphing Quadratic Functions A quadratic function f is given. (a) Express f in standard form. (b) Find the vertex and x- and y-intercepts of f. (c) Sketch a graph of f. (d) Find the domain and range of f. 20. f(x) = 3x2 + 6x 2 Graphing Quadratic Functions A quadratic function f is given. (a) Express f in standard form. (b) Find the vertex and x- and y-intercepts of f. (c) Sketch a graph of f. (d) Find the domain and range of f. 21. f(x) = 2x2 20x + 57 22E 23E 24E 25E Maximum and Minimum Values A quadratic function f is given. (a) Express f in standard form. (b) Sketch a graph of f. (c) Find the maximum or minimum value of f. 26. f(x) = x2 8x + 8 27E 28E Maximum and Minimum Values A quadratic function f is given. (a) Express f in standard form. (b) Sketch a graph of f. (c) Find the maximum or minimum value of f. 29. f(x) = x2 3x + 3 30E 31E 32E Maximum and Minimum Values A quadratic function f is given. (a) Express f in standard form. (b) Sketch a graph of f. (c) Find the maximum or minimum value of f. 33. h(x) = 1 x x2 34E 35E 36E Formula for Maximum and Minimum Values Find the maximum or minimum value of the function. 37. f(t) = 3 + 80t 20t2 38E 39E 40E Formula for Maximum and Minimum Values Find the maximum or minimum value of the function. 41. h(x)=12x2+2x6 42E 43E 44E 45E 46E Finding Quadratic Functions Find a function f whose graph is a parabola with the given vertex and that passes through the given point. 47. Vertex (2, 3); point (3, 1)Finding Quadratic Functions Find a function f whose graph is a parabola with the given vertex and that passes through the given point. 48. Vertex (1, 5); point (3, 7)Maximum of a Fourth-Degree Polynomial Find the maximum value of the function f(x)=3+4x2+x4 [Hint: Let t = x2.]Maximum of a Fourth-Degree Polynomial Find the maximum value of the function f(x)=2+16x3+4x6 [Hint: Let t = x3.]Height of a Ball If a ball is thrown directly upward with a velocity of 40 ft/s, its height (in feet) after t seconds is given by y = 40t 16t2. What is the maximum height attained by the ball?Path of a Ball A ball is thrown across a playing field from a height of 5 ft above the ground at an angle of 45 to the horizontal at a speed of 20 ft/s. It can be deduced from physical principles that the path of the ball is modeled by the function y=32(20)2x2+x+5 where x is the distance in feet that the ball has traveled horizontally. (a) Find the maximum height attained by the ball. (b) Find the horizontal distance the ball has traveled when it hits the ground.Revenue A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R(x) = 80x 0.4x2, where the revenue R(x) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?Sales A soft-drink vendor at a popular beach analyzes his sales records and finds that if he sells x cans of soda pop in one day, his profit (in dollars) is given by P(x)=0.001x2+3x1800 What is his maximum profit per day, and how many cans must he sell for maximum profit?Advertising The effectiveness of a television commercial depends on how many times a viewer watches it. After some experiments an advertising agency found that if the effectiveness E is measured on a scale of 0 to 10, then E(n)=23n190n2 where n is the number of times a viewer watches a given commercial. For a commercial to have maximum effectiveness, how many times should a viewer watch it?Pharmaceuticals When a certain drug is taken orally, the concentration of the drug in the patients bloodstream after t minutes is given by C(t) = 0.06t 0.0002t2, where 0 t 240 and the concentration is measured in mg/L. When is the maximum serum concentration reached, and what is that maximum concentration?Agriculture The number of apples produced by each tree in an apple orchard depends on how densely the trees are planted. If n trees are planted on an acre of land, then each tree produces 900 9n apples. So the number of apples produced per acre is A(n)=n(9009n) How many trees should be planted per acre to obtain the maximum yield of apples?Agriculture At a certain vineyard it is found that each grape vine produces about 10 lb of grapes in a season when about 700 vines are planted per acre. For each additional vine that is planted, the production of each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by A(n)=(700+n)(100.01n) where n is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.Maxima and Minima Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages 237244. 59. Problem 21 21. Fencing a Field Consider the following problem: A farmer has 2400 ft of fencing and wants to fence off a rectangular filed that borders a straight river. He does not need a fence along the river (see the figure). What are the dimensions of the field of largest area that he can fence? (a) Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration, and use your results to estimate the dimensions of the largest possible field. (b) Find a function that models the area of the field in terms of one its sides. (c) Use your model to solve the problem, and compare with your answer to part (a).Maxima and Minima Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages 237244. 60. Problem 22 22. Dividing a Pen A rancher with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle (see the figure). (a) Find a function that models that total area of the four pens. (b) Find the largest possible total area of the four pens.Maxima and Minima Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages 237244. 61. Problem 25 25. Light from a Window A Norman window has the shape of a rectangle surmounted by a semicircle, as shown in the figure to the left. A Norman window with perimeter 30 ft is to be constructed. (a) Find the a function that models the area of the window. (b) Find the dimensions of the window that admits the greatest among of light.Maxima and Minima Use the formulas of this section to give an alternative solution to the indicated problem in Focus on Modeling: Modeling with Functions on pages 237244. 62. Problem 24 24. Maximizing Area A wire 10 cm long is cut into two pieces, one of length x and the other of length 10 x, as shown in the figure. Each piece is bent into the shape of a square. (a) Find a function that models the total area enclosed by the two squares. (b) Find the value of x that minimizes the total area of the two squares.Fencing a Horse Corral Carol has 2400 ft of fencing to fence in a rectangular horse corral. (a) Find a function that models the area of the corral in terms of the width x of the corral. (b) Find the dimensions of the rectangle that maximize the area of the corral.Making a Rain Gutter A rain gutter is formed by bending up the sides of a 30-in.-wide rectangular metal sheet as shown in the figure. (a) Find a function that models the cross-sectional area of the gutter in terms of x. (b) Find the value of x that maximizes the cross-sectional area of the gutter. (c) What is the maximum cross-sectional area for the gutter?Stadium Revenue A baseball team plays in a stadium that holds 55,000 spectators. With the ticket price at 10, the average attendance at recent games has been 27,000. A market survey indicates that for every dollar the ticket price is lowered, attendance increases by 3000. (a) Find a function that models the revenue in terms of ticket price. (b) Find the price that maximizes revenue from ticket sales. (c) What ticket price is so high that no revenue is generated?Maximizing Profit A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost 6, and the society sells an average of 20 per week at a price of 10 each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it will lose 2 sales per week. (a) Find a function that models weekly profit in terms of price per feeder. (b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?67E Only one of the following graphs could be the graph of a polynomial function. Which one? Why are the others not graphs of polynomials?Describe the end behavior of each polynomial. (a) y = x3 8x2 + 2x 15Endbehavior:yasxyasx (b) y = 2x4 + 12x + 100Endbehavior:yasxyasx If c is a zero of the polynomial P, then (a) P(c) = _____. (b) x c is a _____ of P(x). (c) c is a(n) _____ -intercept of the graph of P.Which of the following statements couldnt possibly be true about the polynomial function P? (a) P has degree 3, two local maxima, and two local minima. (b) P has degree 3 and no local maxima or minima. (c) P has degree 4, one local maximum, and no local minima.Transformations of Monomials Sketch the graph of each function by transforming the graph of an appropriate function of the form y = xn from Figure 1. Indicate all x- and y-intercepts on each graph. 5. (a) P(x) = x2 4 (b) Q(x) = (x 4)2 (c) P(x) = 2x2 + 3 (d) P(x) = (x + 2)2 6E 7E 8E End Behavior A polynomial function is given. (a) Describe the end behavior of the polynomial function. (b) Match the polynomial function with one of the graphs IVI. 9. P(x) = x(x2 4)End Behavior A polynomial function is given. (a) Describe the end behavior of the polynomial function. (b) Match the polynomial function with one of the graphs IVI. 10. Q(x) = x2(x2 4)End Behavior A polynomial function is given. (a) Describe the end behavior of the polynomial function. (b) Match the polynomial function with one of the graphs IVI. 11. R(x) = x5 + 5x3 4x End Behavior A polynomial function is given. (a) Describe the end behavior of the polynomial function. (b) Match the polynomial function with one of the graphs IVI. 12. S(x)=12x62x4 End Behavior A polynomial function is given. (a) Describe the end behavior of the polynomial function. (b) Match the polynomial function with one of the graphs IVI. 13. T(x) = x4 + 2x3 End Behavior A polynomial function is given. (a) Describe the end behavior of the polynomial function. (b) Match the polynomial function with one of the graphs IVI. 14. U(x) = x3 + 2x2 Graphing Factored Polynomials Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. 15. P(x) = (x 1)(x + 2)16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E Graphing Polynomials Factor the polynomial and use the factored form to find the zeros. Then sketch the graph. 31. P(x) = x3 x2 6x 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E End Behavior Determine the end behavior of P. Compare the graphs of P and Q in large and small viewing rectangles, as in Example 3(b). 47. P(x) = x4 7x2 + 5x + 5; Q(x) = x4 End Behavior Determine the end behavior of P. Compare the graphs of P and Q in large and small viewing rectangles, as in Example 3(b). 48. P(x) = x5 + 2x2 + x; Q(x) = x5 49E End Behavior Determine the end behavior of P. Compare the graphs of P and Q in large and small viewing rectangles, as in Example 3(b). 50. P(x) = 2x2 x12; Q(x) = x12 51E 52E 53E Local Extrema The graph of a polynomial function is given. From the graph, find (a) the x- and y-intercepts, and (b) the coordinates of all local extrema. 54. P(x)=19x449x3 55E Local Extrema Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. State the domain and range. 56. y = x3 3x2, [2, 5] by [10, 10]57E 58E 59E Local Extrema Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. State the domain and range. 60. y = x4 18x2 + 32, [5, 5] by [100, 100]61E 62E 63E 64E 65E 66E 67E 68E

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