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mathematics
calculus early transcendentals

Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions

A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 16 - x2. What are the dimensions of the rectangle with the maximum area? What is that area?
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.c. For those cases in which the Mean Value Theorem applies, make a sketch of the function and the line that
Evaluate the following limits. 1 – cos 3x lim 8x2
Use Newton’s method to find approximate answers to the following questions.Where are the inflection points of located? 15 x* + 2 f(x) 5 .5 x' + 30x² + 1 3
A rectangular page in a textbook (with width x and length y) has an area of 98 in2, top and bottom margins set at 1 in, and left and right margins set at 1/2 in. The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the
Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error.√146
Determine the following indefinite integrals. Check your work by differentiation.∫(3x5 - 5x9) dx
Find the intervals on which f is increasing and decreasing. Superimpose the graphs of f and f' to verify your work.f(x) = -x4/4 + x3 - x2
a. Find the critical points of the following functions on the domain or on the given interval.b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, local minimum, or neither.f(x) = 3x2 - 4x + 2
Make a complete graph of the following functions. If an interval is not specified, graph the function on its domain. Use a graphing utility to check your work.f(x) = x + 2 cos x on [-2π, 2π]
An 8-ft-tall fence runs parallel to the wall of a house at a distance of 5 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume that the vertical wall of the house is 20 ft high and the horizontal ground extends 20 ft from the fence.
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.c. For those cases in which the Mean Value Theorem applies, make a sketch of the function and the line that
Evaluate the following limits using l’Hôpital’s Rule tan 4z lim z→0 tan 77
Use Newton’s method to find approximate answers to the following questions.Where are all the local extrema of f(x) = 3x4 + 8x3 + 12x2 + 48x located?
Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error.tan 3°
A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the curve y = cos x, for 0 < x < π/2. Approximate the dimensions of the rectangle that maximize the area of the rectangle. What is the area?
Find all the antiderivatives of the following functions. Check your work by taking derivatives.F(t) = π
Find the intervals on which f is increasing and decreasing. Superimpose the graphs of f and f' to verify your work.f(x) = x4 - 4x3 + 4x2
Sketch a graph of a function f continuous on [0, 4] satisfying the given properties.f'(x) = 0 at x = 1 and 3; f'(2) is undefined; f has an absolute maximum at x = 2; f has neither a local maximum nor a local minimum at x = 1; and f has an absolute minimum at x = 3.
Make a complete graph of the following functions. If an interval is not specified, graph the function on its domain. Use a graphing utility to check your work.f(x) = ln (x2 + 1)
A 10-ft-tall fence runs parallel to the wall of a house at a distance of 4 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume the vertical wall of the house and the horizontal ground have infinite extent.
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.c. For those cases in which the Mean Value Theorem applies, make a sketch of the function and the line that
Evaluate the following limits using l’Hôpital’s Rule tan u cot u lim и>п/4 и - п/4
Use Newton’s method to find approximate answers to the following questions.Where is the first local minimum of on the interval (0, ∞) located? cos x f(x)
Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error.1/203
A right triangle has legs of length h and r, and a hypotenuse of length 4 (see figure). It is revolved about the leg of length h to sweep out a right circular cone. What values of h and r maximize the volume of the cone? (Volume of a cone = πr2h/3.) 4
Find all the antiderivatives of the following functions. Check your work by taking derivatives.G(s) = 1/s2 + 1
Find the intervals on which f is increasing and decreasing. Superimpose the graphs of f and f' to verify your work.f(x) = 12 + x - x2
Sketch a graph of a function f continuous on [0, 4] satisfying the given properties.f'(1) and f'(3) are undefined; f'(2) = 0; f has a local maximum at x = 1; f has a local minimum at x = 2; f has an absolute maximum at x = 3; and f has an absolute minimum at x = 4.
Make a complete graph of the following functions. If an interval is not specified, graph the function on its domain. Use a graphing utility to check your work.f(x) = tan-1 x2
A boat on the ocean is 4 mi from the nearest point on a straight shoreline; that point is 6 mi from a restaurant on the shore. A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. a. If she walks at 3 mi/hr and rows at 2 mi/hr, at
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.c. For those cases in which the Mean Value Theorem applies, make a sketch of the function and the line that
Evaluate the following limits using l’Hôpital’s Rule 4т- x sin x + x? – 47? lim х — 2т
Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.y = ln x and y = x3 - 2
a. Write the equation of the line that represents the linear approximation to the following functions at the given point a.b. Graph the function and the linear approximation at a.c. Use the linear approximation to estimate the given function value.d. Compute the percent error in your approximation,
Find all the antiderivatives of the following functions. Check your work by taking derivatives.h(y) = y-1
Find the intervals on which f is increasing and decreasing. Superimpose the graphs of f and f' to verify your work.f(x) = x3 + 4x
Make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.f(x) = x(x - 1) e-x
Sketch a graph of a function f continuous on [0, 4] satisfying the given properties.f'(x) = 0 for x = 1, 2, and 3; f has an absolute minimum at x = 1; f has no local extremum at x = 2; and f has an absolute maximum at x = 3.
Use the guidelines of this section to make a complete graph of f. 4х + 4 .2 Г) x² + 3
Find the point P on the curve y = x2 that is closet to the point (18, 0). What is the least distance between P and (18, 0)?
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.c. For those cases in which the Mean Value Theorem applies, make a sketch of the function and the line that
Evaluate the following limits using l’Hôpital’s Rule 3 sin 4x lim 5x X-
Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.y = 4√x and y = x2 + 1
a. Write the equation of the line that represents the linear approximation to the following functions at the given point a.b. Graph the function and the linear approximation at a.c. Use the linear approximation to estimate the given function value.d. Compute the percent error in your approximation,
Find all the antiderivatives of the following functions. Check your work by taking derivatives.f(x) = ex
Find the intervals on which f is increasing and decreasing. Superimpose the graphs of f and f' to verify your work.f(x) = (x - 1)2
Make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.f(x) = x2/3 + (x + 2)1/3
Sketch a graph of a function f continuous on [0, 4] satisfying the given properties.f'(x) = 0 for x = 1 and 2; f has an absolute maximum at x = 4; f has an absolute minimum at x = 0; and f has a local minimum at x = 2.
Use the guidelines of this section to make a complete graph of f. x² + 12 f(x) 2х + 1
Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)?
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.c. For those cases in which the Mean Value Theorem applies, make a sketch of the function and the line that
Evaluate the following limits using l’Hôpital’s Rule -1 х- 4 tan lim TT х
Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.y = x3 and y = x2 + 1
a. Write the equation of the line that represents the linear approximation to the following functions at the given point a.b. Graph the function and the linear approximation at a.c. Use the linear approximation to estimate the given function value.d. Compute the percent error in your approximation,
Find all the antiderivatives of the following functions. Check your work by taking derivatives.H(z) = -6z-7
Find the intervals on which f is increasing and decreasing. Superimpose the graphs of f and f' to verify your work.f(x) = x2 - 16
Make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. cos TTX on [-2, 2] f(x) 1 + x- .2
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. УА уУ 3 h(x) и а р д rs t
Use the guidelines of this section to make a complete graph of f. 2х - 3 f(x) 2х- 8
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.c. For those cases in which the Mean Value Theorem applies, make a sketch of the function and the line that
A man wishes to get from an initial point on the shore of a circular lake with radius 1 mi to a point on the shore directly opposite (on the other end of the diameter). He plans to swim from the initial point to another point on the shore and then walk along the shore to the terminal point. a.
Evaluate the following limits using l’Hôpital’s Rule In x – 1 lim х>е х — е
Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.y = 1/x and y = 4 - x2
a. Write the equation of the line that represents the linear approximation to the following functions at the given point a.b. Graph the function and the linear approximation at a.c. Use the linear approximation to estimate the given function value.d. Compute the percent error in your approximation,
Find all the antiderivatives of the following functions. Check your work by taking derivatives.f(y) = -2/y3
Find the intervals on which f is increasing and decreasing. Superimpose the graphs of f and f' to verify your work.f(x) = 4 - x2
Make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.f(x) = 3√x - √x + 2
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. у 3 g(х) ь x
Use the guidelines of this section to make a complete graph of f. Зх |f(x) = x² – 1
a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.c. For those cases in which the Mean Value Theorem applies, make a sketch of the function and the line that
A square-based, box-shaped shipping crate is designed to have a volume of 16 ft3. The material used to make the base costs twice as much (per square foot) as the material in the sides, and the material used to make the top costs half as much (per square foot) as the material in the sides. What are
Evaluate the following limits using l’Hôpital’s Rule et – 1 lim х>0 х* + Зх
Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.y = ex and y = x3
a. Write the equation of the line that represents the linear approximation to the following functions at the given point a.b. Graph the function and the linear approximation at a.c. Use the linear approximation to estimate the given function value.d. Compute the percent error in your approximation,
Use analytical methods to evaluate the following limits. 1 + 2 + ·.. + n lim ,2 n² n 00 п(п + 1) Use 1 + 2 + · ·.+ n =
Find all the antiderivatives of the following functions. Check your work by taking derivatives.Q(s) = csc2 s
The following figures give the graph of the derivative of a continuous function f that passes through the origin. Sketch a possible graph of f on the same set of axes. y = f'(x) х -1
Make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. x? + x f(x) : 4 – x2
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. y = f(x) ардг S ь x
Use the guidelines of this section to make a complete graph of f. x2 f(x) = x² – 4
The fastest drag racers can reach a speed of 330 mi/hr over a quarter-mile strip in 4.45 seconds (from a standing start). Complete the following sentence about such a drag racer: At some point during the race, the maximum acceleration of the drag racer is at least _______ mi/hr/s.
Suppose an airline policy states that all baggage must be box-shaped with a sum of length, width, and height not exceeding 108 in. What are the dimensions and volume of a square based box with the greatest volume under these conditions?
Evaluate the following limits using l’Hôpital’s Rule In x lim x→1 4x – x² – 3
Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.y = sin x and y = x/2
a. Write the equation of the line that represents the linear approximation to the following functions at the given point a.b. Graph the function and the linear approximation at a.c. Use the linear approximation to estimate the given function value.d. Compute the percent error in your approximation,
Find all the antiderivatives of the following functions. Check your work by taking derivatives.P(x)= 3 sec2 x
The following figures give the graph of the derivative of a continuous function f that passes through the origin. Sketch a possible graph of f on the same set of axes. УА y = f'(x) х 1
Make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.f(x) = 4 cos (π(x - 1)) on [0, 2]
Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. УА y = f(x) +
Use the guidelines of this section to make a complete graph of f. х* f(x) х — 2
Concurrent measurements indicate that at an elevation of 6.1 km, the temperature is -10.3°C, and at an elevation of 3.2 km, the temperature is 8.0°C. Based on the Mean Value Theorem, can you conclude that the lapse rate exceeds the threshold value of 7°C/km at some intermediate elevation?
Of all boxes with a square base and a volume of 100 m3, which one has the minimum surface area? (Give its dimensions.)
Evaluate the following limits using l’Hôpital’s Rule x4 + x' + 2x + 2 lim
a. Write the equation of the line that represents the linear approximation to the following functions at the given point a.b. Graph the function and the linear approximation at a.c. Use the linear approximation to estimate the given function value.d. Compute the percent error in your approximation,
Use a calculator or program to compute the first 10 iterations of Newton’s method when it is applied to the following functions with the given initial approximation. Make a table similar to that in Example 1.f(x) = ln (x + 1) - 1; x0 = 1.7
Find all the antiderivatives of the following functions. Check your work by taking derivatives.g(x) = -4 cos 4x
Sketch a graph of a function that is continuous on (-∞, ∞) and has the following properties. Use a sign graph to summarize information about the function.f'(-2) = f'(2) = f'(6) = 0; f'(x) ≥ 0 on (-∞, ∞)
Make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. Зх f(x) х x² + 3
Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum value or an absolute minimum value. y = g(x) + х

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