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mathematics
applied calculus

Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions

If f(x) and g(x) are differentiable functions such that f(2) = f′(2) = 3, g(2) = 3, and g′(2) = 1/3, compute the following derivatives: d dx - [x(g(x) - f(x))] |x=2
Refer to Exercise 61.(a) FindGive an interpretation for these values.(b) Use the chain rule to findExercise 61After a computer software company went public, the price of one share of its stock fluctuated according to the graph in Fig. 1(a). The total worth of the company depended on the value of
Refer to Exercise 61.(a) What was the maximum value of the company during the first 6 months since it went public, and when was that maximum value attained?(b) Assuming that the value of one share will continue to increase at the rate that it did during the period following the 6th month, what is
The body mass index, or BMI, is a ratio of a person’s weight divided by the square of his or her height. Let b (t) denote the BMI; then,where t is the age of the person, w(t) the weight in kilograms, and h(t) the height in meters. Find an expression for b′(t). b(t) = [h(t)1²¹
Let f(x) = 1/x and g(x) = x3.(a) Show that the product rule yields the correct derivative of (1/x)x3 = x2.(b) Compute the product f′(x)g′(x), and note that it is not the derivative of f (x)g (x).
Apply the special case of the general power ruleand the identityto prove the product rule. d dx - [h(x)]²= 2h(x)h'(x) =
The derivative of (x3 - 4x)/x is obviously 2x for x ≠ 0, because (x3 - 4x)/x = x2 - 4 for x ≠ 0. Verify that the quotient rule gives the same derivative.
Let f(x), g (x), and h(x) be differentiable functions. Find a formula for the derivative of f (x)g (x)h (x).
In an expression of the form f(g(x)), f(x) is called the outer function and g(x) is called the inner function. Give a written description of the chain rule using the words inner and outer.
The BMI is usually used as a guideline to determine whether a person is overweight or underweight. For example, according to the Centers for Disease Control, a 12-year-old boy is at risk of being overweight if his BMI is between 21 and 24 and is considered overweight if his BMI is above 24. A
State as many terms used to describe graphs of functions as you can recall.
Refer to the functions whose graphs are given in Fig. 17.Which functions have a positive first derivative for all x? V (d) Figure 17 x X Y y (b) X x y Y (c) (f) X X
Find the x-intercepts of the given function.y = x2 - 3x + 1
Sketch the graphs of the following functions.f(x) = x3 - 6x2 + 12x - 6
Some years ago, it was estimated that the demand for steel approximately satisfied the equation p = 256 - 50x, and the total cost of producing x units of steel was C(x) = 182 + 56x. (The quantity x was measured in millions of tons and the price and total cost were measured in millions of dollars.)
Refer to graphs (a)–(f) in Fig. 19.(a)(b)(c)(d)(e)(f)Which functions are increasing for all x? y X
Figure 6 shows the inventory levels of dried Rainier cherries at a natural food store in Seattle and the order–reorder periods over 1 year. Refer to the figure to answer the following questions.(a) What is the average amount of cherries in inventory during one order–reorder period?(b) What is
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 the graph of f(x) concave up? Concave down? Figure 1 2 2 Y y =
Each of the graphs of the functions in Exercises 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.f(x) = x3 - 27x
Refer to the functions whose graphs are given in Fig. 17.Which functions have a negative first derivative for all x? V (d) Figure 17 x X Y y (b) X x y Y (c) (f) X X
For what x does the function g(x) = 10 + 40x - x2 have its maximum value?
Refer to graphs (a)–(f) in Fig. 19.(a)(b)(c)(d)(e)(f)Which functions are decreasing for all x? y X
Refer to Fig. 6. Suppose that (i) the ordering cost for each delivery of dried cherries is $50, and (ii) it costs $7 to carry 1 pound of dried cherries in inventory for 1 year.(a) What is the inventory cost (carrying cost plus ordering cost) if carrying costs are computed on the average inventory
Refer to the functions whose graphs are given in Fig. 17.Which functions have a positive second derivative for all x? V (d) Figure 17 x X Y y (b) X x y Y (c) (f) X X
Given the cost function C(x) = x3 - 6x2 + 13x + 15, find the minimum marginal cost.
Find the x-intercepts of the given function.y = x2 + 5x + 5
What is the difference between having a relative maximum at x = 2 and having an absolute maximum at x = 2?
Each of the graphs of the functions in Exercises 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.f(x) = x3 - 6x2 + 1
Refer to graphs (a)–(f) in Fig. 19.(a)(b)(c)(d)(e)(f)Which functions have the property that the slope always increases as x increases? y X
Find the maximum value of the function f(x) = 12x - x2, and give the value of x where this maximum occurs.
The revenue function for a oneproduct firm isFind the value of x that results in maximum revenue. R(x) = 200 1600 x + 8 X.
Refer to the functions whose graphs are given in Fig. 17.Which functions have a negative second derivative for all x? V (d) Figure 17 x X Y y (b) X x y Y (c) (f) X X
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.
Find the x-intercepts of the given function.y = 2x2 + 5x + 2
Give three characterizations of what it means for the graph of f (x) to be concave up at x = 2. Concave down.
Each of the graphs of the functions in Exercises 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. ƒ(x) = −6x³ − ³x² + 3x − 3 - -
Each of the graphs of the functions in Exercises 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.f(x) = -x3 + 6x2 - 9x + 1
A pharmacist wants to establish an optimal inventory policy for a new antibiotic that requires refrigeration in storage. The pharmacist expects to sell 800 packages of this antibiotic at a steady rate during the next year. She plans to place several orders of the same size spaced equally throughout
Find the minimum value of f(t) = t3 - 6t2 + 40, t ≥ 0, and give the value of t where this minimum occurs.
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) and f′(x) increasing
Find the x-intercepts of the given function.y = 4 - 2x - x2
Refer to graphs (a)–(f) in Fig. 19.(a)(b)(c)(d)(e)(f)Which functions have the property that the slope always decreases as x increases? y X
What does it mean to say that the graph of f (x) has an inflection point at x = 2?
Which one of the graphs in Fig. 18 could represent a function f(x) for which f(a) > 0, f′(a) = 0, and f″(a) < 0? y Figure 18 a (a) a (c) X X fi a (b) a (d) x X
A furniture store expects to sell 640 sofas at a steady rate next year. The manager of the store plans to order these sofas from the manufacturer by placing several orders of the same size spaced equally throughout the year. The ordering cost for each delivery is $160, and carrying costs, based on
For what t does the function f (t) = t2 - 24t have its minimum value?
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) and f′(x) decreasing
Each of the graphs of the functions in Exercises 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. I EX E [ + zx − x ³ = (x)ƒ
Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously. 1 y X
The revenue function for a particular product is R(x) = x(4 - .0001x). Find the largest possible revenue.
Find the x-intercepts of the given function.y = 4x - 4x2 - 1
Each of the graphs of the functions in Exercises 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. f(x)= x³ = x + 2 -
What is the difference between an x-intercept and a zero of a function?
Which one of the graphs in Fig. 18 could represent a function f(x) for which f(a) = 0, f′(a) < 0, and f″(a) > 0? y Figure 18 a (a) a (c) X X fi a (b) a (d) x X
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 cost of placing an order with the
Find the maximum of Q = xy if x + y = 2.
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) increasing and f′(x) decreasing
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.
Find the x-intercepts of the given function.y = 3x2 + 10x + 3
Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously. y 10 H X
How do you determine the y-intercept of a function?
Exercises refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property.f′(x) is positive. y Figure 3 a b I c d e y = f(x) X
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 average number of tires in storage, amount to
Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously. 1 I
Find two positive numbers x and y that maximize Q = x2y if x + y = 2.
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
A small tie shop sells ties for $3.50 each. The daily cost function is estimated to be C(x) dollars, where x is the number of ties sold on a typical day and C(x) = .0006x3 - .03x2 + 2x + 20. Find the value of x that will maximize the store’s daily profit.
The demand equation for a certain commodity is0 ≤ x ≤ 60. Find the value of x and the corresponding price p that maximize the revenue. P 1 12 - 10x + 300,
Sketch the graph of a function that has the properties described.f(2) = 1; f′(2) = 0; concave up for all x.
What is an asymptote? Give an example.
Each of the graphs of the functions in Exercises 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.f(x) = -x3 - 12x2 - 2
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 of microscopes in the warehouse, amount to
Find the minimum of Q = x2 + y2 if x + y = 6.
Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously. 1 hi
Exercises refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property.f′(x) is negative. y Figure 3 a b I c d e y = f(x) X
Sketch the graph of a function that has the properties described.f (-1) = 0; f′(x) < 0 for x < -1, f′(-1) = 0 and f′(x) > 0 for x > -1.
State the first-derivative rule. The second-derivative rule.
Each of the graphs of the functions in Exercises 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.f(x) = 2x3 + 3x2 - 3
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 interest payments) is $2 per book, to be
In Exercise, can there be a maximum for Q = x2 + y2 if x + y = 6? Justify your answer.
The demand equation for a product is p = 2 - .001x. Find the value of x and the corresponding price, p, that maximize the revenue.
Sketch the graph of a function that has the properties described.f(3) = 5; f′(x) > 0 for x < 3, f′(3) = 0 and f′(x) > 0 for x > 3.
Exercises refer to the graph in Fig. 3. List the labeled values of x at which the derivative has the stated property.f″(x) is positive. y Figure 3 a b I c d e y = f(x) X
Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously. 1 y = x p
Give two connections between the graphs of f (x) and f′(x).
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
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 will be s dollars per item, to be figured on
Use the given information to make a good sketch of the function f(x) near x = 3. f(3) = 4, f'(3) = -1/1, ƒ"(3) = 5
Find the positive values of x and y that minimize S = x + y if xy = 36, and find this minimum value.
Describe the way the slope changes as you move along the graph (from left to right) in Exercise 5.Describe each of the following graphs. Your descriptions should include each of the six categories mentioned previously. 1 y X
Properties of various functions are described next. In each case,draw some conclusion about the graph of the function. ƒ(1) = 2, ƒ'(1) > 0
Sketch the graphs of the following functions.f(x) = 5 - 13x + 6x2 - x3
What is an objective equation?
Use the given information to make a good sketch of the function f(x) near x = 3. f(3) = -2, f'(3) = 0, ƒ"(3) = 1
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
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. Label each edge of the square end with the
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. How many prints should the artist make to
Sketch the graphs of the following functions.f(x) = 2x3 + x - 2
What is a constraint equation?
Sketch the following parabolas. Include their x- and y-intercepts.y = 4 + 3x - x2

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