Property of functions is described next. Draw some conclusion about the graph of the function. f (4) = -2, f ‘(4) > 0, f ’’(4) = -1
> Figure 1 contains the graph of f ‘(x), the derivative of f (x). Use the graph to answer the following questions about the graph of f (x). (a) For what values of x is the graph of f (x) increasing? Decreasing? (b) For what values of x is
> 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
> 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
> 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.
> 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 re
> 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 12 bushel for each additional tree that is planted. How many trees should be planted
> 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? Figure
> 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 di
> 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.
> The tangent line to the curve y = x3 - 6x2 - 34x - 9 has slope 2 at two points on the curve. Find the two points.
> Find the minimum value of the function g(t) = t2 - 6t + 9, 1 ≤ t ≤ 6.
> 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.
> For what x does the function f (x) = 1/4 x2 - x + 2, 0 ≤ x ≤ 8, have its maximum value?
> Let f (x) be the number of people living within x miles of the center of New York City. (a) What does f (10 + h) - f (10) represent? (b) Explain why f ‘(10) cannot be negative.
> 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
> 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. Information A. s(t) is a constant function. B. s’(
> Let f (x) be a function whose derivative is f ‘(x) = √(5x2 + 1). Show that the graph of f (x) has an inflection point at x = 0.
> Let f (x) be a function whose derivative is f ‘(x) = 1/(1 + x2). Note that f ‘(x) is always positive. Show that the graph of f (x) has an inflection point at x = 0.
> Show that the function f (x) = (2x2 + 3) 3/2 is decreasing for x < 0 and increasing for x > 0.
> Let f (x) = (x2 + 2)3/2. Show that the graph of f (x) has a possible relative extreme point at x = 0.
> The tangent line to the curve y = 1/3 x3 - 4x2 + 18x + 22 is parallel to the line 6x - 2y = 1 at two points on the curve. Find the two points.
> Sketch the following curves. y = 1/2x + 2x + 1 (x > 0)
> Sketch the following curves. y = x/5 + 20/x + 3 (x > 0)
> Sketch the following curves. y = x4 - 4x3
> Sketch the following curves. y = x4 - 2x2
> Sketch the following curves. y = x3 - 6x2 - 15x + 50
> Sketch the following curves. y = - 1/3 x3 - 2x2 - 5x
> Sketch the following curves. y = x3 - 3x2 - 9x + 7
> Sketch the following curves. y = 11/3 + 3x - x2 – 1/3 x3
> Sketch the following curves. y = 100 + 36x - 6x2 - x3
> Sketch the following curves. y = x3 - 3x2 + 3x - 2
> If h(x) = [f (x)]2 + 1/g(x), determine h(1) and h’(1), given that f (1) = 1, g(1) = 4, f ‘(1) = -1, and g’(1) = 4.
> Sketch the following curves. y = x3 – 3/2 x2 - 6x
> Sketch the following curves. y = 2x3 + 3x2 + 1
> Sketch the following parabolas. Include their x- and y-intercepts. y = 2x2 + x - 1
> Sketch the following parabolas. Include their x- and y-intercepts. y = -x2 + 20x - 90
> Sketch the following parabolas. Include their x- and y-intercepts. y = -x2 + 8x - 13
> Sketch the following parabolas. Include their x- and y-intercepts. y = x2 + 3x + 2
> Sketch the following parabolas. Include their x- and y-intercepts. y = x2 - 9x + 19
> Sketch the following parabolas. Include their x- and y-intercepts. y = -2x2 + 10x - 10
> Sketch the following parabolas. Include their x- and y-intercepts. y = 4 + 3x - x2
> Sketch the following parabolas. Include their x- and y-intercepts. y = x2 + 3x - 10
> If g(1) = 4 and g’(1) = 3, find f (1) and f ‘(1), where f (x) = 5 * √g(x).
> Sketch the following parabolas. Include their x- and y-intercepts. y = 7 + 6x - x2
> Sketch the following parabolas. Include their x- and y-intercepts. y = 3 - x2
> 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 195
> 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? Figure 4: 2.4 2.0 1.6 1.2 .8 4 fi 4 y =
> Property of functions is described next. Draw some conclusion about the graph of the function. H(0) = 0, H’(0) = 0, H’’(0) = 1
> Property of functions is described next. Draw some conclusion about the graph of the function. g(5) = -1, g’(5) = -2, g’’(5) = 0
> Property of functions is described next. Draw some conclusion about the graph of the function. G (10) = 2, G’(10) = 0, G’’(10) > 0
> Property of functions is described next. Draw some conclusion about the graph of the function. F ‘(2) = -1, F ’’(2) 6 0
> Property of functions is described next. Draw some conclusion about the graph of the function. h’(3) = 4, h’’(3) = 1
> If g(3) = 2 and g’(3) = 4, find f (3) and f ‘(3), where f (x) = 2 * [g(x)]3.
> Property of functions is described next. Draw some conclusion about the graph of the function. g(1) = 5, g’(1) = -1
> Property of functions is described next. Draw some conclusion about the graph of the function. f (1) = 2, f ‘(1) > 0
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’ (x) is minimized. y a b c d e y = f(x)
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’ (x) is maximized. y a b c d e y = f(x)
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’’(x) is negative. y a b c d e y = f(x)
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’’ (x) is positive. y a b c d e y = f(x)
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’(x) is negative. y a b c d e y = f(x)
> Refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property. Figure 3: f ’(x) is positive. y a b c d e y = f(x)
> Draw the graph of a function f (x) for which the function and its first derivative have the stated property for all x. f (x) decreasing and f (x) increasing
> What is the difference between an x-intercept and a zero of a function?
> If f (5) = 2, f ‘(5) = 3, g(5) = 4, and g’(5) = 1, find h(5) and h’(5), where h(x) = 3f (x) + 2g(x).
> What does it mean to say that the graph of f (x) has an inflection point at x = 2?
> Give three characterizations of what it means for the graph of f (x) to be concave up at x = 2. Concave down.
> What is the difference between having a relative maximum at x = 2 and having an absolute maximum at x = 2?
> State as many terms used to describe graphs of functions as you can recall.
> How are the cost, revenue, and profit functions related?
> Outline the procedure for solving an optimization problem.
> What is a constraint equation?
> What is an objective equation?
> Outline a procedure for sketching the graph of a function.
> Outline a method for locating the inflection points of a function.
> Draw the graph of f (x) = 2x + 18/x - 10 in the window [0, 16] by [0, 16]. In what ways is this graph like the graph of a parabola that opens upward? In what ways is it different?
> Outline a method for locating the relative extreme points of a function.
> Give two connections between the graphs of f (x) and f (x).
> State the first-derivative rule. The second-derivative rule.
> What is an asymptote? Give an example.
> How do you determine the y-intercept of a function?
> A one-product firm estimates that its daily total cost function (in suitable units) is C(x) = x3 - 6x2 + 13x + 15 and its total revenue function is R(x) = 28x. Find the value of x that maximizes the daily profit.
> The revenue function for a particular product is R(x) = x (4 - .0001x). Find the largest possible revenue.
> The revenue function for a one-product firm is R(x) = 200 – 1600/(x + 8) - x. Find the value of x that results in maximum revenue.
> If a total cost function is C(x) = .0001x3 - .06x2 + 12x + 100, is the marginal cost increasing, decreasing, or not changing at x = 100? Find the minimum marginal cost.
> Given the cost function C(x) = x3 - 6x2 + 13x + 15, find the minimum marginal cost.
> Figure 3 contains the curves y = f (x), y = g(x), and y = h(x) and the tangent lines to y = f (x) and y = g(x) at x = 1, with h(x) = f (x) + g(x). Find h(1) and h’(1). Figure 3: Y y = y = y = h(x) y = f(x) - 4x + 2.6 26x + 1.1 Fig
> Differentiate. y = x/2 – 2/x
> The cost function for a manufacturer is C(x) dollars, where x is the number of units of goods produced and C, C , and C are the functions given in Fig. 15. Figure 15: (a) What is the cost of manufacturing 60 units of goods? (b) What is the marginal c
> 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). Figure 14: Y 80 70 60 50 40 30 20 10 Y 3.2 2.4 1.6 .8 -.8 -1.6 -2.4 -
> 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 th
> 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 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
> 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 co
> A certain toll road averages 36,000 cars per day when charging $1 per car. A survey concludes that increasing the toll will result in 300 fewer cars for each cent of increase. What toll should be charged to maximize the revenue?
> 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
> A swimming club offers memberships at the rate of $200, provided that a minimum of 100 people join. For each member in excess of 100, the membership fee will be reduced $1 per person (for each member). At most, 160 memberships will be sold. How many memb
> An artist is planning to sell signed prints of her latest work. If 50 prints are offered for sale, she can charge $400 each. However, if she makes more than 50 prints, she must lower the price of all the prints by $5 for each print in excess of the 50. H
> Draw the graph of f (x) = 1/6 x3 – 5/2 x2 + 13x - 20 in the window [0, 10] by [-20, 30]. Algebraically determine the coordinates of the inflection point. Zoom in and zoom out to convince yourself that there are no relative extreme points anywhere.
> The average ticket price for a concert at the opera house was $50. The average attendance was 4000. When the ticket price was raised to $52, attendance declined to an average of 3800 persons per performance. What should the ticket price be to maximize th
> Until recently hamburgers at the city sports arena cost $4 each. The food concessionaire sold an average of 10,000 hamburgers on a game night. When the price was raised to $4.40, hamburger sales dropped off to an average of 8000 per night. (a) Assuming a
