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mathematics
applied calculus

Questions and Answers of Applied Calculus

In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x2 - y2 = 1
Differentiate the functions in Exercises. y = (2x² = x + 1)(-x³ + 1) -
Differentiate the functions in Exercises. y = (x² + x + 1)²(x - 1)4
State the chain rule. Give an example.
Compute f(g(x)), where f(x) and g(x) are the following: f(x) = x + 1 X-3' g(x) = x + 3
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x3 + y3 - 6 = 0
What is the relationship between the chain rule and the general power rule?
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.y5 - 3x2 = x
Differentiate the functions in Exercises. y = x(x² + 1)4
Differentiate the functions in Exercises. y=xVx
What does it mean for a function to be defined implicitly by an equation?
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x4 + (y + 3)4 = x2
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.y4 - x4 = y2 - x2
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x).h(x) = (x3 + 8x - 2)5
Outline the procedure for solving a related-rates problem.
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x). h(x) = V4-x²
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x3 + y3 = x2 + y2
Differentiate the functions in Exercises. y = (x² + 3)(x² - 3)10
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x).h(x) = (9x2 + 2x - 5)7
Differentiate the functions in Exercises. y=[(-2x³ + x)(6x - 3)]4
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x). h(x) = 1 r3 – 5,2+1
Differentiate the functions in Exercises. y = x²(3x4 + 12x - 1)²
Differentiate the functions in Exercises. y=(5r+1)(x2 –1)+ 2x + 1 3
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.2x3 + y = 2y3 + x
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x). h(x) = (4x - 3)³ + 1 4x 3 -
Differentiate the functions in Exercises. y= x-1 x + 1
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x4 + 4y = x - 4y3
Each of the following functions may be viewed as a composite function h(x) = f ( g (x)). Find f(x) and g(x).h(x) = (5x2 + 1)-1/2
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.xy = 5
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.xy3 = 2
Differentiate the functions in Exercises. y 1 x² + x + 7 2
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = (x2 + 5)15
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x(y + 2)5 = 8
Differentiate the functions in Exercises. y = - 1 2 x² + 1 X
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = (x4 + x2)10
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x2y3 = 6
Differentiate the functions in Exercises. y X x +
Let f(x) = (3x + 1)4(3 - x)5. Find all x such that f′(x) = 0.
Differentiate the functions in Exercises. y x + 3 (2x + 1)²
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = 6x2(x - 1)3
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x3y2 - 4x2 = 1
Let f(x) = (x2 + 1)/(x2 + 5). Find all x such that f′(x) = 0.
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = 5x3(2 - x)4
Find the equation of the line tangent to the graph ofat the point where x = 0. y= x-3 √4+x²
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.(x + 1)2 (y - 1)2 = 1
Find the equation of the line tangent to the graph of y = (x3 - 1)(x2 + 1)4 at the point where x = -1.
A botanical display is to be constructed as a rectangular region with a river as one side and a sidewalk 2 meters wide along the inside edges of the other three sides. (See Fig. 1.) The area for the
Differentiate the functions in Exercises. y || 1 | TT + 2 x² + 1
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = 2(x3 - 1)(3x2 + 1)4
Given f(1) = 1, f′(1) = 5, g(1) = 3, g′(1) = 4, f′(3) = 2, and g′(3) = 6, compute the following derivatives: d dx [f(g(x)) g(x))] x=1
Differentiate the functions in Exercises. 4 y = x² +4√x
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x3 + y3 = x3y3
Differentiate the functions in Exercises using one or more of the differentiation rules discussed thus far.y = 2(2x - 1)5/4(2x + 1)3/4
Repeat Exercise 17, with the sidewalk on the inside of all four sides. In this case, the 800-square-meter planted region has dimensions x - 4 meters by y - 4 meters.Exercise 17A botanical display is
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x2 + 4xy + 4y = 1
Given f(1) = 1, f′(1) = 5, g(1) = 3, g′(1) = 4, f′(3) = 2, and g′(3) = 6, compute the following derivatives: [=x ОЛА xp P
Given f(1) = 1, f′(1) = 5, g(1) = 3, g′(1) = 4, f′(3) = 2, and g′(3) = 6, compute the following derivatives: d | dx [g (f(x))] x=1
Differentiate the functions in Exercises. y 1-2 (x² + 1)²
Differentiate the functions in Exercises. y= ax + b cx + d
Given f(1) = 1, f′(1) = 5, g(1) = 3, g′(1) = 4, f′(3) = 2, and g′(3) = 6, compute the following derivatives: d dx [g(g(x))] x=1
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x2y + y2x = 3
In Exercises, suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx.x3y + xy3 = 4
A store estimates that its cost when selling x lamps per day is C dollars, where C = 40x + 30 (the marginal cost per lamp is $40). If daily sales are rising at the rate of three lamps per day, how
Differentiate the functions in Exercises. (x + 1)³ 2 (x - 5)²
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point.4y3 - x2 = -5; x = 3, y = 1
Differentiate the functions in Exercises. y = [(3x² + 2x + 2)(x - 2)]²
A company pays y dollars in taxes when its annual profit is P dollars. If y is some (differentiable) function of P and P is some function of time t, give a chain rule formula for the time rate of
Differentiate the functions in Exercises. y 1 V √x - 2
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point. xy³ = 2; x = − 1, y = −2
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point.y2 = x3 + 1; x = 2, y = -3
Differentiate the functions in Exercises. y 1 √x + 1
In Exercises, find a formula for d/dx f(g (x)), where f (x) is a function such that f′(x) = 1/(x2 + 1).g(x) = x3
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x).h(x) = f(x2)
In Exercises, find a formula for d/dx f(g (x)), where f (x) is a function such that f′(x) = 1/(x2 + 1).g(x) = 1/x
Differentiate the functions in Exercises. y = 3 √x+1
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x).h(x) = 2 f(2x + 1)
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point.√x + √y = 7; x = 9, y = 16
In Exercises, find a formula for d/dx f(g (x)), where f (x) is a function such that f′(x) = 1/(x2 + 1).g(x) = x2 + 1
Differentiate the functions in Exercises. y = 3 (x + 11) ³ - 3
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)
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x). h(x) = f(x²)
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point.xy + y3 = 14; x = 3, y = 2
In Exercises, find a formula for d/dx f(g (x)), where f(x) is a function such that f′(x) = x√1 - x2.g(x) = x2
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x). h(x) = √f(x²)
In Exercises, a function h(x) is defined in terms of a differentiable f(x). Find an expression for h′(x).h(x) = f(f (x))
Use implicit differentiation of the equations in Exercises to determine the slope of the graph at the given point.y2 = 3xy - 5; x = 2, y = 1
In Exercises, find a formula for d/dx f(g (x)), where f(x) is a function such that f′(x) = x√1 - x2.g(x) = √x
Find the equation of the tangent line to the graph of x2 y4 = 1 at the point (4, 1/2) and at the point (4, - 1/2).
In Exercises, find a formula for d/dx f(g (x)), where f(x) is a function such that f′(x) = x√1 - x2.g(x) = x3/2
The amount, A, of anesthetics that a certain hospital uses each week is a function of the number, S, of surgical operations performed each week. Also, S, in turn, is a function of the population, P,
Differentiate the functions in Exercises. y = x + 3 2 x² + 1
Differentiate the functions in Exercises. y = √3x - 1 X
Differentiate the functions in Exercises. y = √x + 2(2x + 1)²
In Exercises, find dy/dx, where y is a function of u such thatState the answer in terms of x only.u = x2 + 1 dy U du +1 u²
Find the equation of the tangent line to the graph of x4 y2 = 144 at the point (2, 3) and at the point (2, -3).
Suppose that x and y represent the amounts of two basic inputs for a production process and that the equationdescribes all input amounts where the output of the process is 1080 units(a) Find
In Exercises, find dy/dx, where y is a function of u such thatState the answer in terms of x only. dy U du +1 u²
Sketch the graph of y = 4x/(x + 1)2, x > -1.
In Exercises, find dy/dx, where y is a function of u such thatu = x2 dy du U V1 + u
Compute d/dx f(g (x)), where f(x) and g(x) are the following: f(x)=√x, g(x) = x² + 1

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