Find the quadratic function f (x) = ax2 + bx + c that goes through (0, 1) and has a local minimum at (1,-1).
> Refer to the inventory problem of Example 2. If the distributor offers a discount of $1 per case for orders of 600 or more cases, should the manager change the quantity ordered?
> A store manager wants to establish an optimal inventory policy for an item. Sales are expected to be at a steady rate and should total Q items sold during the year. Each time an order is placed a cost of h dollars is incurred. Carrying costs for the year
> A bookstore is attempting to determine the most economical order quantity for a popular book. The store sells 8000 copies of this book per year. The store figures that it costs $40 to process each new order for books. The carrying cost (due primarily to
> Foggy Optics, Inc., makes laboratory microscopes. Setting up each production run costs $2500. Insurance costs, based on the average number of microscopes in the warehouse, amount to $20 per microscope per year. Storage costs, based on the maximum number
> The Great American Tire Co. expects to sell 600,000 tires of a particular size and grade during the next year. Sales tend to be roughly the same from month to month. Setting up each production run costs the company $15,000. Carrying costs, based on the a
> A California distributor of sporting equipment expects to sell 10,000 cases of tennis balls during the coming year at a steady rate. Yearly carrying costs (to be computed on the average number of cases in stock during the year) are $10 per case, and the
> For what t does the function f (t) = t2 - 24t have its minimum value?
> Find the minimum value of f (t) = t3 - 6t2 + 40, t Ú 0, and give the value of t where this minimum occurs.
> Find the maximum value of the function f (x) = 12x - x2, and give the value of x where this maximum occurs.
> Differentiate the function f (x) = (3x2 + x - 2)2 in two ways. (a) Use the general power rule. (b) Multiply 3x2 + x - 2 by itself and then differentiate the resulting polynomial
> For what x does the function g(x) = 10 + 40x - x2 have its maximum value?
> Find the value of x for which the rectangle inscribed in the semicircle of radius 3 in Fig. 19 has the greatest area. Figure 19: -3 ม (r, y) 3
> Find the point on the line y = -2x + 5 that is closest to the origin. City A 6 miles Road X Road 11 miles City B 11-x 4 miles
> Find the point on the graph of y = √x that is closest to the point (2, 0). See Fig. 17. Figure 17: 2 0 Y 1 2 3
> A rectangular page is to contain 50 square inches of print. The page has to have a 1-inch margin on top and at the bottom and a 12-inch margin on each side (see Fig. 16). Find the dimensions of the page that minimize the amount of paper used. NIL in
> A cable is to be installed from one corner, C, of a rectangular factory to a machine, M, on the floor. The cable will run along one edge of the floor from C to a point, P, at a cost of $6 per meter, and then from P to M in a straight line buried under th
> A ship uses 5x2 dollars of fuel per hour when traveling at a speed of x miles per hour. The other expenses of operating the ship amount to $2000 per hour. What speed minimizes the cost of a 500-mile trip?
> In Example 3 we can solve the constraint equation (2) for x instead of w to get x = 20 – ½ w. Substituting this for x in (1), we get Sketch the graph of the equation and show that the maximum occurs when w = 20 and x =
> A large soup can is to be designed so that the can will hold 16p cubic inches (about 28 ounces) of soup. [See Fig. 14(b).] Find the values of x and h for which the amount of metal needed is as small as possible. Figure 14: x 2x (a) h x h (b) 2m x S
> Figure 14(a) shows a Norman window, which consists of a rectangle capped by a semicircular region. Find the value of x such that the perimeter of the window will be 14 feet and the area of the window will be as large as possible. Figure 14: 2x (a)
> Find the equation of the tangent line to the curve y = 8 / x2 + x + 2 at x = 2.
> Find two positive numbers, x and y, whose product is 100 and whose sum is as small as possible.
> Find two positive numbers, x and y, whose sum is 100 and whose product is as large as possible.
> Find the dimensions of the rectangular garden of greatest area that can be fenced off (all four sides) with 300 meters of fencing.
> 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
> 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.
> 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.
> 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 dim
> 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.
> Consider the problem of finding the dimensions of the rectangular garden of area 100 square meters for which the amount of fencing needed to surround the garden is as small as possible. (a) Draw a picture of a rectangle and select appropriate letters for
> Postal requirements specify that parcels must have length plus girth of at most 84 inches. Consider the problem of finding the dimensions of the square-ended rectangular package of greatest volume that is mailable. (a) Draw a square-ended rectangular box
> Find the slope of the tangent line to the curve y = (x2 - 15)6 at x = 4. Then write the equation of this tangent line.
> Figure 12(b) shows an open rectangular box with a square base. Consider the problem of finding the values of x and h for which the volume is 32 cubic feet and the total surface area of the box is minimal. (The surface area is the sum of the areas of the
> There are $320 available to fence in a rectangular garden. The fencing for the side of the garden facing the road costs $6 per foot and the fencing for the other three sides costs $2 per foot. [See Fig. 12(a).] Consider the problem of finding the dimensi
> Find the positive values of x, y, and z that maximize Q = xyz, if x + y = 1 and y + z = 2. What is this maximum value?
> Find the positive values of x and y that minimize S = x + y if xy = 36, and find this minimum value.
> In Exercise 7, can there be a maximum for Q = x2 + y2 if x + y = 6? Justify your answer. Exercise 7: Find the minimum of Q = x2 + y2 if x + y = 6.
> Find the minimum of Q = x2 + y2 if x + y = 6.
> Find two positive numbers x and y that maximize Q = x2y if x + y = 2.
> Find the maximum of Q = xy if x + y = 2.
> Find the x-intercepts of the given function. y = 4 - 2x - x2
> Find the x-intercepts of the given function. y = 2x2 + 5x + 2
> Write the equation of the tangent line to the curve y = x3 + 3x - 8 at (2, 6).
> Find the x-intercepts of the given function. y = x2 + 5x + 5
> Find the x-intercepts of the given function. y = x2 - 3x + 1
> The canopy height (in meters) of the tropical bunch-grass elephant millet t days after mowing (for t Ú 32) is f (t) = -3.14 + .142t - .0016t2 + .0000079t3 - .0000000133t4. (Source: Crop Science.) (a) Graph f (t) in the window [32, 250] by [-1.2, 4.5]. (b
> In a medical experiment, the body weight of a baby rat in the control group after t days was f (t) = 4.96 + .48t + .17t2 - .0048t3 grams. (Source: Growth, Development and Aging.) (a) Graph f (t) in the window [0, 20] by [-12, 50]. (b) Approximately how m
> If f (a) = 0 and f (x) is decreasing at x = a, explain why f (x) must have a local maximum at x = a.
> If f (a) = 0 and f (x) is increasing at x = a, explain why f (x) must have a local minimum at x = a.
> Find the quadratic function f (x) = ax2 + bx + c that goes through (2, 0) and has a local maximum at (0, 1).
> Determine which function is the derivative of the other. f(x) g(x) y X
> Determine which function is the derivative of the other. f(x) |(x)b y
> Find the slope of the tangent line to the curve y = x3 + 3x – 8 at (2, 6).
> Sketch the graphs of the following functions for x > 0. y = 1/√x + x/2
> Sketch the graphs of the following functions for x > 0. y = 6√x - x
> Sketch the graphs of the following functions for x > 0. y = 1/x2 + x/4 – 5/4
> Sketch the graphs of the following functions for x > 0. y = 2/x +x/2 + 2
> Sketch the graphs of the following functions for x > 0. y = 12/x + 3x + 1
> Sketch the graphs of the following functions for x > 0. y = 9/x + x + 1
> Sketch the graphs of the following functions for x > 0. y = 2/x
> Sketch the graphs of the following functions for x > 0. y = 1/x + ¼ x
> Sketch the graphs of the following functions. f (x) = (x + 2)4 - 1
> Sketch the graphs of the following functions. f (x) = (x - 3)4
> Find the slope of the graph of y = f (x) at the designated point. f (x) = x10 + 1 + √(1 – x), (0, 2)
> Differentiate. y = 1/3x3
> Sketch the graphs of the following functions. f (x) = 3x4 - 6x2 + 3
> Sketch the graphs of the following functions. f (x) = x4 - 6x2
> Sketch the graphs of the following functions. f (x) = 1/3 x3 - 2x2
> Sketch the graphs of the following functions. f (x) = 1 - 3x + 3x2 - x3
> Sketch the graphs of the following functions. f (x) = -3x3 - 6x2 - 9x - 6
> Sketch the graphs of the following functions. f (x) = 4/3 x3 - 2x2 + x
> Sketch the graphs of the following functions. f (x) = 2x3 + x - 2
> Sketch the graphs of the following functions. f (x) = 5 - 13x + 6x2 - x3
> Sketch the graphs of the following functions. f (x) = x3 + 2x2 + 4x
> Sketch the graphs of the following functions. f (x) = x3 + 3x + 1
> Find the slope of the graph of y = f (x) at the designated point. f (x) = 3x2 - 2x + 1, (1, 2)
> Sketch the graphs of the following functions. f (x) = -x3
> Sketch the graphs of the following functions. f (x) = x3 - 6x2 + 12x - 6
> Show that the function f (x) = -x3 + 2x2 - 6x + 3 is always decreasing.
> Show that the function f (x) = 1/3 x3 - 2x2 + 5x has no relative extreme points.
> Find the x-intercepts of the given function. y = 3x2 + 10x + 3
> Find the x-intercepts of the given function. y = 4x - 4x2 - 1
> The graph of the function has one relative maximum and one relative minimum point. Find these points using the first-derivative test. Use a variation chart as in Example 1. Variation Chart from Example 1: f (x) = x3 - 6x2 +1 Critical Values Interv
> The graph of the function has one relative maximum and one relative minimum point. Find these points using the first-derivative test. Use a variation chart as in Example 1. Variation Chart from Example 1: f (x) = x3 - 27x Critical Values Intervals
> Index-Fund Fees Suppose that the cost function in Exercise 45 is C(x) = -2.5x + 1, where x% is the index-fund fee. (The company has a fixed cost of $1 billion, and the cost decreases as a function of the index-fund fee.) Find the value of x that maximize
> 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 com
> Differentiate. y = (x – 1/x)-1
> 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(
> 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? Figure 17: y Figure 17 2 y = g(x) x
> Determine which function is the derivative of the other. x- (x)b = h (x)f=h fi
> Determine which function is the derivative of the other. fi y = f(x) y = g(x)
> The graph of function has one relative extreme point. Find it (giving both x- and y-coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. f (x) = 5x2 + x - 3
> The graph of function has one relative extreme point. Find it (giving both x- and y-coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. g(x) = x2 + 10x + 10
> The graph of function has one relative extreme point. Find it (giving both x- and y-coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. g(x) = 3 + 4x - 2x2
> The graph of function has one relative extreme point. Find it (giving both x- and y-coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. f (x) = 5 - 12x - 2x2
> The graph of function has one relative extreme point. Find it (giving both x- and y-coordinates) and determine if it is a relative maximum or a relative minimum point. Do not include a sketch of the graph of the function. f (x) = 1/4 x2 - 2x + 7
> Let a, b, c, d be fixed numbers with a ≠ 0, and let f (x) = ax3 + bx2 + cx + d. Is it possible for the graph of f (x) to have more than one inflection point? Explain your answer.
> Differentiate. f (x) = (√x/2 + 1)3/2
> Let a, b, c be fixed numbers with a ≠ 0, and let f (x) = ax2 + bx + c. Is it possible for the graph of f (x) to have an inflection point? Explain your answer.
> Sketch the following curves, indicating all relative extreme points and inflection points. y = x4 – 4/3 x3
