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mathematics
precalculus

Calculus Of A Single Variable 11th Edition Ron Larson, Bruce H. Edwards - Solutions

Evaluate the definite integral using the values below. -6 "6 [~ ³ dx = 320, [²x dx = 16, [*dx = 4 2 2
Evaluate the definite integral. Use a graphing utility to verify your result. n/3 -π/3 4 sec 0 tane de
Find the indefinite integral and check the result by differentiation.∫(sin x − 6 cos x) dx
Find the indefinite integral and check the result by differentiation. ∫(csc x cot x − 2x) dx
Find the indefinite integral and check the result by differentiation.∫(θ2 + sec2 θ) dθ
Sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral (a > 0, r > 0).∫r-r √r2 - x2 dx
In your own words, describe primary equation, secondary equation, and feasible domain.
What is the equation of the tangent line approximation to the graph of a function f at the point (c, f(c))?
Calculate two iterations of Newton’s Method to approximate a zero of the function using the given initial guess.f(x) = x2 − 5, x1 = 2
Why does Newton’s Method fail when f′(xn) = 0? What does this mean graphically?
In your own words, describe the guidelines for solving applied minimum and maximum problems.
What do the differentials of x and y mean?
Find the absolute extrema of the function on the closed interval.f(x) = x3 + 6x2, [−6, 1]
Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not.f(x) = x2/3, [1, 8] f'(c) = f(b)-f(a) b-a
Find the absolute extrema of the function on the closed interval.f(x) = √x − 2, [0, 4]
Explain how to find a differential of a function.
Use the information to find and compare Δy and dy. Function y = 0.5x³ x-Value x = 1 Differential of x Ax = dx = 0.1
Find the absolute extrema of the function on the closed interval.h(x) = x − 3√x, [0, 9]
Find two positive numbers that satisfy the given requirements.The sum is S and the product is a maximum.
Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not.f(x) = 2x − 3√x, [−1, 1] f'(c)
Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not.f(x) = 1/x, [1, 4] f'(c) = f(b)-f(a) b-a
Use Newton’s Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results.f(x) = 2 − x3
Use the information to find and compare Δy and dy. Function y = x - 2r³ x-Value x=3 Differential of x Ax = dx = 0.001
Find two positive numbers that satisfy the given requirements.The sum of the first number and twice the second number is 108 and the product is a maximum.
Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not.f(x) = ∣5 − x∣, [2, 6] f'(c)
Find two positive numbers that satisfy the given requirements.The sum of the first number cubed and the second number is 500 and the product is a maximum.
Determine whether Rolle’s Theorem can be applied to f on the closed interval [a, b]. If Rolle’s Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = 0. If Rolle’s Theorem cannot be applied, explain why not.f(x) = (x − 2)(x + 3)2 , [−3, 2]
Use Newton’s Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results.f(x) = 5√x − 1 − 2x
Determine whether Rolle’s Theorem can be applied to f on the closed interval [a, b]. If Rolle’s Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = 0. If Rolle’s Theorem cannot be applied, explain why not.f(x) = x2/1 − x2, [−2, 2]
Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not. f'(c) = f(b)-f(a) b-a
Determine whether Rolle’s Theorem can be applied to f on the closed interval [a, b]. If Rolle’s Theorem can be applied, find all values of c in the open interval (a, b) such that f′(c) = 0. If Rolle’s Theorem cannot be applied, explain why not.f(x) = sin 2x, [− π, π]
Find the points on the graph of the function that are closest to the given point.y = x2, (0, 3)
Find the length and width of a rectangle that has the given area and a minimum perimeter.Area: 49 square feet
Use the information to find and compare Δy and dy. Function y = 7x² - 5x x-Value x = -4 Differential of x Ax = dx = 0.001
Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values of c in the open interval (a, b) such thatIf the Mean Value Theorem cannot be applied, explain why not.f(x) = √x − 2x, [0, 4] f'(c)
Find the points on the graph of the function that are closest to the given point.y = x2 − 2, (0, −1)
A rectangular poster is to contain 648 square inches of print. The margins at the top and bottom of the poster are to be 2 inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the poster be so that the least amount of poster is used?
Find the points of inflection and discuss the concavity of the graph of the function.f(x) = sin x/2, [0, 4π]
Find the open intervals on which the function is increasing or decreasing.h(x) = √x(x − 3), x > 0
Consider the function f(x) = x3 − 3x2 + 3.(a) Use a graphing utility to graph f.(b) Use Newton’s Method to approximate a zero with x1 = 1 as the initial guess.(c) Repeat part (b) using x1 = 1/4 as the initial guess and observe that the result is different.(d) To understand why the results in
Find the differential dy of the given function.y = 3x2/3
Find the open intervals on which the function is increasing or decreasing.f(x) = (x − 1)2 (2x − 5)
Does Newton’s Method fail when the initial guess is a relative maximum of f ? Explain.
What will be the values of future guesses for x if your initial guess is a zero of f? Explain.
(a) Find the critical numbers of f, if any,(b) Find the open intervals on which the function is increasing or decreasing,(c) Apply the First Derivative Test to identify all relative extrema, (d) Use a graphing utility to confirm your results.f(x) = 4x3 − 5x
(a) Find the critical numbers of f, if any,(b) Find the open intervals on which the function is increasing or decreasing,(c) Apply the First Derivative Test to identify all relative extremaf(t) = 1/4 t4 − 8t
(a) Find the critical numbers of f, if any,(b) Find the open intervals on which the function is increasing or decreasing,(c) Apply the First Derivative Test to identify all relative extrema, and(d) Use a graphing utility to confirm your results.f(x) = x3 − 8x/4
(a) Find the critical numbers of f, if any,(b) Find the open intervals on which the function is increasing or decreasing,(c) Apply the First Derivative Test to identify all relative extremaf(x) = x + 4/x2
(a) Find the critical numbers of f, if any,(b) Find the open intervals on which the function is increasing or decreasing,(c) Apply the First Derivative Test to identify all relative extrema.f(x) = cos x − sin x, (0, 2π)
(a) Find the critical numbers of f, if any,(b) Find the open intervals on which the function is increasing or decreasing,(c) Apply the First Derivative Test to identify all relative extrema, and(d) Use a graphing utility to confirm your results. (x) = sin(-1). 3 (0,4)
The range R of a projectile iswhere v0 is the initial velocity in feet per second and is the angle of elevation. Use differentials to approximate the change in the range when v0 = 2500 feet per second and is changed from 10° to 11°. R= 32 (sin 2)
The graphs of f, f′, and f ″ are shown on the same set of coordinate axes. Identify each graph. Explain your reasoning. To print an enlarged copy of the graph, go to MathGraphs.com. -2 -2 y 2 Xx
The cross sections of an irrigation canal are isosceles trapezoids of which three sides are 8 feet long (see figure). Determine the angle of elevation of the sides such that the area of the cross sections is a maximum by completing the following. (a) Analytically complete six rows of a table such
(a) Find the critical numbers of f, if any,(b) Find the open intervals on which the function is increasing or decreasing,(c) Apply the First Derivative Test to identify all relative extrema, and(d) Use a graphing utility to confirm your results.f(x) = x2 − 3x − 4/x − 2
Sketch the graph of f(x) = 2 − 2 sin x on the interval [0, π/2]. (a) Find the distance from the origin to the y-intercept and the distance from the origin to the x-intercept.(b) Write the distance d from the origin to a point on the graph of f as a function of x.(c) Use calculus to find the
Outlays for national defense D (in billions of dollars) for 2006 through 2014 are shown in the table, where t is the time in years, with t = 6 corresponding to 2006.(a) Use the regression capabilities of a graphing utility to find a model of the form D = at4 + bt3 + ct2 + dt + e for the data.(b)
The function s(t) describes the motion of a particle along a line.(a) Find the velocity function of the particle at any time t ≥ 0.(b) Identify the time interval(s) on which the particle is moving in a positive direction.(c) Identify the time interval(s) on which the particle is moving in a
Use the graph of f′ to sketch a graph of f and the graph of f″. To print an enlarged copy of the graph, go to MathGraphs.com. + -4-3 432 1 f + 34 X
The function s(t) describes the motion of a particle along a line.(a) Find the velocity function of the particle at any time t ≥ 0.(b) Identify the time interval(s) on which the particle is moving in a positive direction.(c) Identify the time interval(s) on which the particle is moving in a
The graphs of f, f′, and f ″ are shown on the same set of coordinate axes. Identify each graph. Explain your reasoning. To print an enlarged copy of the graph, go to MathGraphs.com. -4 -2 4 -4 y 2 4 X
Find the points of inflection and discuss the concavity of the graph of the function.f(x) = x3 − 9x2
Find the points of inflection and discuss the concavity of the graph of the function.f(x) = 6x4 − x2
The period of a pendulum is given by T = 2 √L/g where L is the length of the pendulum in feet, g is the acceleration due to gravity, and T is the time in seconds. The pendulum has been subjected to an increase in temperature such that the length has increased by 1/2%. (a) Find the
Consider a symmetric cross inscribed in a circle of radius r (see figure).(a) Write the area A of the cross as a function of x and find the value of x that maximizes the area.(b) Write the area A of the cross as a function of and find the value of that maximizes the area.(c) Show that the critical
Find all relative extrema of the function. Use the Second Derivative Test where applicable.g(x) = 2x2 (1 − x2)
Use the graph of f′ to sketch a graph of f and the graph of f″. To print an enlarged copy of the graph, go to MathGraphs.com. -9 -6 3 2 y -2 -3 + f 36 X
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Newton’s Method fails when the initial guess x1 corresponds to a horizontal tangent line for the graph of f at x1.
The graph of a function f is shown below. To print an enlarged copy of the graph, go to MathGraphs.com. (a) Sketch f′.(b) Use the graph to estimate lim x→∞ f(x) and lim x→∞ f′(x).(c) Explain the answers you gave in part (b). + -4 -2 6 4 2 -2 y f 2 4 x
Find all relative extrema of the function. Use the Second Derivative Test where applicable.f(x) = x4 − 2x2 + 6
Use the graph of f′ to sketch a graph of f and the graph of f″. To print an enlarged copy of the graph, go to MathGraphs.com. 20 16 12 00 8 4 -8-4 y 4 + -X 8 12 16
The manager of a store recorded the annual sales S (in thousands of dollars) of a product over a period of 7 years, as shown in the table, where t is the time in years, with t = 8 corresponding to 2008. (a) Use the regression capabilities of a graphing utility to find a model of the form S = at3 +
Use a graphing utility to graph the function and identify any horizontal asymptotes. f(x) = X x² + 6
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If y = x + c, then dy = dx
Sketch the graph of a function f having the given characteristics.f(0) = 4, f(6) = 0f′(x) < 0 for x < 2 or x > 4f′(2) does not exist. f′(4) = 0f′(x) > 0 for 2 < x < 4f″(x) < 0 for x ≠ 2
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If y = ax + b, then ∆y/∆x = dy/dx.
The graph of f is shown in the figure.(a) For which values of x is f′(x) zero? Positive? Negative? What do these values mean?(b) For which values of x is f″(x) zero? Positive? Negative? What do these values mean?(c) On what open interval is f′ an increasing function?(d) For which value of x
Use the graph of f′ to sketch a graph of f and the graph of f″. To print an enlarged copy of the graph, go to MathGraphs.com. 3 -02 1 -3 -2 -1 -3 y 1 2 3
Find the point on the graph of the equation 16x = y2 that is closest to the point (6, 0).
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The tangent line approximation at any point for any linear equation is the linear equation itself.
Use a graphing utility to graph the function and identify any horizontal asymptotes. f(x) /4x² - 1 8x + 1
Find the point on the graph of the function x = √10y that is closest to the point (0, 4).
The graph of the first derivative of a function f on the interval [−7, 5] is shown. Use the graph to answer each question.(a) On what interval(s) is f decreasing?(b) On what interval(s) is the graph of f concave downward?(c) At what x-value(s) does f have relative extrema?(d) At what
Use a graphing utility to graph the function and identify any horizontal asymptotes.f(x) = 3/x + 4
The graph of the first derivative of a function f on the interval [−4, 2] is shown. Use the graph to answer each question.(a) On what interval(s) is f increasing?(b) On what interval(s) is the graph of f concave upward?(c) At what x-value(s) does f have relative extrema?(d) At what
Consider the function f(x) = tan(sin πx).(a) Use a graphing utility to graph the function.(b) Identify any symmetry of the graph.(c) Is the function periodic? If so, what is the period?(d) Identify any extrema on (−1, 1).(e) Use a graphing utility to determine the concavity of the
Analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.f(x) = x√16 − x2
Use the information to find and compare Δy and dy. Function y = 4x³ x-Value x = 2 Differential of x Ax = dx = 0.1
Analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.f(x) = x1/3 (x + 3)2/3
Use a graphing utility to graph the function and determine the slant asymptote of the graph analytically. Zoom out repeatedly and describe how the graph on the display appears to change. Why does this occur?f(x) = x3 − 3x2 + 2/x(x − 3)
Analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.f(x) = x4 − 2x2 + 6
Use the information to find and compare Δy and dy. Function y = r – 5x x-Value x = -3 Differential of .x Ax = dx = 0.01
Use a graphing utility to graph the function and determine the slant asymptote of the graph analytically. Zoom out repeatedly and describe how the graph on the display appears to change. Why does this occur?f(x) = −x3 − 2x2 + 2/2x2
Find two positive numbers such that the sum of twice the first number and three times the second number is 216 and the product is a mzaximum.
Find the point on the graph of f(x) = √x that is closest to the point (6, 0).
Use Newton’s Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results.f(x) = x3 − 3x − 1
Use Newton’s Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results.f(x) = x3 + 2x + 1
Use Newton’s Method to approximate the zero(s) of the function. Continue the iterations until two successive approximations differ by less than 0.001. Then find the zero(s) using a graphing utility and compare the results.f(x) = 3√x − 1 − x
Find the differential dy of the given function.y = x(1 − cos x)
Use differentials to approximate the value of the expression. Compare your answer with that of a calculator.√63.9
Find the differential dy of the given function.y = √36 − x2

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