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mathematics
applied calculus

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

Find the equation of the tangent line to the curve y = (x - 2)5 (x + 1)2 at the point (3, 16).
Compute d/dx f(g (x)), where f(x) and g(x) are the following: zx= [= (x)8 X I (x) f
Compute d/dx f(g (x)), where f(x) and g(x) are the following: f(x)= = 1 1 + √x' g(x)= = - X
Find the equation of the tangent line to the curve y = (x + 1)/(x - 1) at the point (2, 3).
In Exercises, find dy/dx, where y is a function of u such that dy du U V1 + u
Exercises refer to the graphs of the functions f(x) and g (x) in Fig. 2. Determine h(1) and h′(1). 3 Q1 Y 0 y = f(x) CO X
Suppose that x and y represent the amounts of two basic inputs for a production process and 10x1/2y1/2 = 600.Find dy/dx when x = 50 and y = 72.
Compute d/dx f(g (x)), where f(x) and g(x) are the following: ħ − zx = (x) 8 ³zx — fx = (x)ƒ x², x² 4
Find the inflection points on the graph of y = 1 2 x² + 1 X
Find all x-coordinates of points (x, y) on the curve y = (x - 2)5/(x - 4)3 where the tangent line is horizontal.
Exercises refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h′(1). 3 Q1 Y 0 y = f(x) CO X
In Exercises, suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt.x4 + y4 = 1
The graph of y = (x2 - 1)4 (x2 + 1)5 is shown in Fig. 3. Find the coordinates of the local maxima and minima. -1 Figure 3 Y 1| y = (x² - 1)(x² + 1)5 1 X
Exercises refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h′(1). 3 Q1 Y 0 y = f(x) CO X
In Exercises, suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt.y4 - x2 = 1
Compute d/dx f(g (x)), where f(x) and g(x) are the following: f(x) 4 ·+ x², g(x) = 1 - x4 X
Find all x such that dy/dx = 0, where y = (x2 - 4)3 (2x2 + 5)5.
In Exercises, suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt.3xy - 3x2 = 4
Exercises refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h′(1). 3 Q1 Y 0 y = f(x) CO X
In Exercises, suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt.y2 = 8 + xy
Compute d/dx f(g (x)), where f(x) and g(x) are the following:f(x) = (x3 + 1)2, g (x) = x2 + 5
Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. y = Vu+1, u = 2x²
Exercises refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h′(1). 3 Q1 Y 0 y = f(x) CO X
Find the point(s) on the graph of y = (x2 + 3x - 1)/x where the slope is 5.
Exercises refer to the graphs of the functions f (x) and g (x) in Fig. 2. Determine h(1) and h′(1). 3 Q1 Y 0 y = f(x) CO X
In Exercises, suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt.x2 + 2xy = y3
Compute d/dx f(g (x)), where f(x) and g(x) are the following:f(x) = x(x - 2)4, g(x) = x3
Find the point(s) on the graph of y = (2x4 + 1)(x - 5) where the slope is 1.
In Exercises, suppose that x and y are both differentiable functions of t and are related by the given equation. Use implicit differentiation with respect to t to determine dy/dt in terms of x, y, and dx/dt.x2y2 = 2y3 + 1
Find d2y/dx2. y=xVx+1
Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only.y = u3/2, u = 4x + 1
Find d2y/dx2.y = (x2 + 1)4
A point is moving along the graph of x2 - 4y2 = 9. When the point is at (5, -2), its x-coordinate is increasing at the rate of 3 units per second. How fast is the y-coordinate changing at that moment?
Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. - x2 11 zx − x = n - + ; 11 2 y = 2
The revenue, R, that a company receives is a function of the weekly sales, x. Also, the sales level, x, is a function of the weekly advertising expenditures, A, and A, in turn, is a varying function of time.(a) Write the derivative symbols for the following quantities: (i) rate of change of
Find d2y/dx2.
Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. y = u² + 2u u + 1 -, u = x(x + 1)
Find d2y/dx2. y = 2 2+x²
A point is moving along the graph of x3y2 = 200. When the point is at (2, 5), its x-coordinate is changing at the rate of -4 units per minute. How fast is the y-coordinate changing at that moment?
The monthly advertising revenue, A, and the monthly circulation, x, of a magazine are related approximately by the equationwhere A is given in thousands of dollars and x is measured in thousands of copies sold. At what rate is the advertising revenue changing if the current circulation is x = 25
Computey = x2 - 3x, x = t2 + 3, t0 = 0 dy dt = to
Suppose that the price p (in dollars) and the weekly sales x (in thousands of units) of a certain commodity satisfy the demand equation 2p3 + x2 = 4500. Determine the rate at which sales are changing at a time when x = 50, p = 10, and the price is falling at the rate of $.50 per week.
The graph of x2/3 + y2/3 = 8 is the astroid in Fig. 3.(a) Find dy/dx by implicit differentiation.(b) Find the slope of the tangent line at (8, -8). y Figure 3 Astroid. 22/3 + y2/3 = 8 X
The graph of x3 + y3 = 9xy is the folium of Descartes shown in Fig. 4.(a) Find dy/dx by implicit differentiation.(b) Find the slope of the curve at (2, 4). x + y + 1 = 0 Asymptote y -X x² + y² = 9xy Figure 4 Folium of Descartes.
Compute.y(x2 - 2x + 4)2, x = 1/(t + 1), t0 = 1 dy dt = to
Suppose that the price p (in dollars) and the weekly demand, x (in thousands of units) of a commodity satisfy the demand equation 6p + x + xp = 94. How fast is the demand changing at a time when x = 4, p = 9, and the price is rising at the rate of $2 per week?
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x). h(x) = f(x) x²+1
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x).h(x) = xf(x)
Compute.y = x + 1/x - 1, x = t2/4, t0 = 3 dy dt = to
Compute.y = √x + 1, x = √t + 1, t0 = 0 dy dt = to
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x). h(x) = = (f(x)) 2
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x).h(x) = (x2 + 2x - 1) f(x)
Figure 7 shows a 10-foot ladder leaning against a wall.(a) Use the Pythagorean theorem to find an equation relating x and y.(b) If the foot of the ladder is being pulled along the ground at the rate of 3 feet per second, how fast is the top end of the ladder sliding down the wall at the time when
Suppose that in Boston the wholesale price, p, of oranges (in dollars per crate) and the daily supply, x (in thousands of crates), are related by the equation px + 7x + 8p = 328. If there are 4 thousand crates available today at a price of $25 per crate, and if the supply is changing at the rate of
A baseball diamond is a 90-foot by 90-foot square. (See Fig. 9.) A player runs from first to second base at the speed of 22 feet per second. How fast is the player’s distance from third base changing when he is halfway between first and second base? Third base 90 90 Second base X Home Figure 9 A
An airplane flying 390 feet per second at an altitude of 5000 feet flew directly over an observer. Figure 8 shows the relationship of the airplane to the observer at a later time.(a) Find an equation relating x and y.(b) Find the value of x when y is 13,000.(c) How fast is the distance from the
In Exercises, x and y are related by the given equation. Use implicit differentiation to calculate the value of dy/dx for the given values of x and y.x2y2 = 9; x = 1, y = 3
A motorcyclist is driving over a ramp as shown in Fig. 10 at the speed of 80 miles per hour. How fast is she rising? Figure 10 x 1000 ft h 100 ft
In Exercises, x and y are related by the given equation. Use implicit differentiation to calculate the value of dy/dx for the given values of x and y.xy4 = 48; x = 3, y = 2
Find the equation of the line tangent to the graph ofat the point (1, 1). y = X √2-x²
Find the equation of the line tangent to the graph of y = 2x(x - 4)6 at the point (5, 10).
An open rectangular box is 3 feet long and has a surface area of 16 square feet. Find the dimensions of the box for which the volume is as large as possible.
In Exercises, x and y are related by the given equation. Use implicit differentiation to calculate the value of dy/dx for the given values of x and y.x2 - xy3 = 20; x = 5, y = 1
A closed rectangular box is to be constructed with one side 1 meter long. The material for the top costs $20 per square meter, and the material for the sides and bottom costs $10 per square meter. Find the dimensions of the box with the largest possible volume that can be built at a cost of $240
The functionhas one relative minimum point for x ≥ 0. Find it. f(x)=√x² - 6x + 10
In Exercises, x and y are related by the given equation. Use implicit differentiation to calculate the value of dy/dx for the given values of x and y.xy2 - x3 = 10; x = 2, y = 3
A town library estimates that, when the population is x thousand persons, approximately y thousand books will be checked out of the library during 1 year, where x and y are related by the equation y3 - 8000x2 = 0.(a) Use implicit differentiation to find a formula for dy/dx, the rate of change of
Find the x-coordinates of all points on the curve y = (-x2 + 4x - 3)3 with a horizontal tangent line.
A sugar refinery can produce x tons of sugar per week at a weekly cost of .1x2 + 5x + 2250 dollars. Find the level of production for which the average cost is at a minimum and show that the average cost equals the marginal cost at that level of production.
A factory’s weekly production costs y and its weekly production quantity x are related by the equation y2 - 5x3 = 4, where y is in thousands of dollars and x is in thousands of units of output.(a) Use implicit differentiation to find a formula for dy/dx, the marginal cost of production.(b) Find
A cigar manufacturer produces x cases of cigars per day at a daily cost of 50x(x + 200)/(x + 100) dollars. Show that his cost increases and his average cost decreases as the output x increases.
Suppose that P, y, and t are variables, where P is a function of y and y is a function of t.(a) Write the derivative symbols for the following quantities:(i) The rate of change of y with respect to t; (ii) The rate of change of P with respect to y; (iii) The rate of change of P with
A manufacturer plans to decrease the amount of sulfur dioxide escaping from its smokestacks. The estimated cost–benefit function iswhere f(x) is the cost in millions of dollars for eliminating x% of the total sulfur dioxide. (See Fig. 4.) Find the value of x at which the rate of increase of the
The length, x, of the edge of a cube is increasing.(a) Write the chain rule for dV/dt, the time rate of change of the volume of the cube.(b) For what value of x is dV/dt equal to 12 times the rate of increase of x?
Let R(x) be the revenue received from the sale of x units of a product. The average revenue per unit is defined by AR = R(x)/x. Show that at the level of production where the average revenue is maximized, the average revenue equals the marginal revenue.
Suppose that the price p and quantity x of a certain commodity satisfy the demand equation 6p + 5x + xp = 50 and that p and x are functions of time, t. Determine the rate at which the quantity x is changing when x = 4, p = 3, and dp/dt = -2.
Suppose that Q, x, and y are variables, where Q is a function of x and x is a function of y.(a) Write the derivative symbols for the following quantities:(i) The rate of change of x with respect to y; (ii) The rate of change of Q with respect to y; (iii) The rate of change of Q with
Many relations in biology are expressed by power functions, known as allometric equations, of the form y = kxa, where k and a are constants. For example, the weight of a male hognose snake is approximately 446x3 grams, where x is its length in meters. If a snake has length .4 meters and is growing
An offshore oil well is leaking oil onto the ocean surface, forming a circular oil slick about .005 meter thick. If the radius of the slick is r meters, the volume of oil spilled is V = .005πr2 cubic meters. If the oil is leaking at a constant rate of 20 cubic meters per hour so that dV/dt = 20,
The width of a rectangle is increasing at a rate of 3 inches per second and its length is increasing at the rate of 4 inches per second. At what rate is the area of the rectangle increasing when its width is 5 inches and its length is 6 inches?
Animal physiologists have determined experimentally that the weight W (in kilograms) and the surface area S (in square meters) of a typical horse are related by the empirical equation S = 0.1W2/3. How fast is the surface area of a horse increasing at a time when the horse weighs 350 kg and is
Suppose that a kitchen appliance company’s monthly sales and advertising expenses are approximately related by the equation xy - 6x + 20y = 0, where x is thousands of dollars spent on advertising and y is thousands of dishwashers sold. Currently, the company is spending 10 thousand dollars on
Use the fact that at the beginning of 1998, the population of the United States was 268,924,000 people and growing at the rate of 1,856,000 people per year.At the beginning of 1998, the annual consumption of ice cream in the United States was 12,582,000 pints and growing at the rate of 212 million
When a company produces and sells x thousand units per week, its total weekly profit is P thousand dollars, whereThe production level at t weeks from the present is x = 4 + 2t.(a) Find the marginal profit, dP/dx.(b) Find the time rate of change of profit, dP/dt.(c) How fast (with respect to time)
Use the fact that at the beginning of 1998, the population of the United States was 268,924,000 people and growing at the rate of 1,856,000 people per year.At the beginning of 1998, the annual per capita consumption of gasoline in the United States was 52.3 gallons and growing at the rate of .2
The cost of manufacturing x cases of cereal is C dollars, where C = 3x + 4√x + 2. Weekly production at t weeks from the present is estimated to be x = 6200 + 100t cases.(a) Find the marginal cost, dC/dx.(b) Find the time rate of change of cost, dC/dt.(c) How fast (with respect to time) are costs
If f(x) and g(x) are differentiable functions, find g(x) if you know that d − f(g(x)) = 3x² • ƒ'(x³ + 1). dx
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 [f(x)g (x)] x=2
Figure 6 shows the graph offor 0 ≤ x ≤ 2. Find the coordinates of the minimum point. y 1 2 + 2 1² - 2x + 1 x² - 2x + 2 2 X
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: f(x) dx g(x). x=2
If f (x) and g (x) are differentiable functions, such that f(1) = 2, f′(1) = 3, f′(5) = 4, g(1) = 5, g′(1) = 6, g′(2) = 7, and g'(5) = 8, find d dx -f(g(x)) x=1
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: P dx [(f(x) x=2
Ecologists estimate that, when the population of a certain city is x thousand persons, the average level L of carbon monoxide in the air above the city will be L ppm (parts per million), where L = 10 + .4x + .0001x2. The population of the city is estimated to be x = 752 + 23t + .5t2 thousand
If f (x) and g(x) are differentiable functions, find g(x) if you know that f′(x) = 1/x and d dx -f(g(x)) = 2x + 5 2 r+5r - 4
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 [(g(x))²] | x=2
A manufacturer of microcomputers estimates that t months from now it will sell x thousand units of its main line of microcomputers per month, where x = .05t2 + 2t + 5. Because of economies of scale, the profit P from manufacturing and selling x thousand units is estimated to be P = .001x2 + .1x -
Consider the functions of Exercise 59. FindExercise 59If f (x) and g (x) are differentiable functions, such that f(1) = 2, f′(1) = 3, f′(5) = 4, g(1) = 5, g′(1) = 6, g′(2) = 7, d =1² dx g(f(x)) x=1
Refer to Exercise 61. Use the chain rule to findGive an interpretation for these values.Exercise 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 one share and
After 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 one share and was estimated to be(a) Find the total value of the company when t = 1.5 and when t = 3.5.(b)
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 [xf(x)] x=2

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