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mathematics
applied calculus

Questions and Answers of Applied Calculus

Sketch the graph of y = 2/(1 + x2).
In Exercises, find dy/dx, where y is a function of u such thatu = √x dy du U V1 + u
Compute d/dx f(g (x)), where f(x) and g(x) are the following:f(x) = x5, g (x) = 6x - 1
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,
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,
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,
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,
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,
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,
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
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
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
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 =
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 =
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
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
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
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
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.
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
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
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)
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
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%
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
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
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
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
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
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
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
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.
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
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
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
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
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,
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
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,
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,

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