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mathematics
calculus early transcendentals

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

Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates.100x; xx
Rank the functions x3, ln x, xx, and 2x in order of increasing growth rates as x→∞?
Suppose you want to travel D miles at a constant speed of (60 + x) mi/hr, where x could be positive or negative. The time in minutes required to travel D miles is T(x) = 60D(60 + x)-1.Use the result of Exercise 9 to approximate the amount of time it takes to drive 80 miles at 57 mi/hr. What is the
Find all the antiderivatives of the following functions. Check your work by taking derivatives.f(x) = 5x4
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) = sin x + x - 1; x0 = 1.5
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'(x) < 0 on (-∞, 2); f'(x) > 0 on (2, 5); f'(x) < 0 on (5, ∞)
Find the solution of the following initial value problems.p'(t) = 10e-t; p(0) = 100
A man is stands 1 mi east of a crossroads. At noon, a dog starts walking north from the crossroads at 1 mi/hr (see figure). At the same instant, the man starts walking and at all times walks directly toward the dog at s > 1 mi/hr. The path in the xy plane followed by the man as he pursues his
Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.f(x) = 6x2 - x3
a. Find the critical points of the following functions on the given interval.b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.c. Find the absolute maximum and minimum values on the given interval when they exist.f(t) = 3t/(t2 +
Determine the following indefinite integrals. [x+ – 2Vx + 2 dx x2
Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates.ln x; ln (ln x)
Find the solution of the following initial value problems.f'(u) = 4(cos u - sin 2u); f(π/6) = 0
The figure shows six containers, each of which is filled from the top. Assume that water is poured into the containers at a constant rate and each container is filled in 10 seconds. Assume also that the horizontal cross sections of the containers are always circles. Let h(t) be the depth of water
Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.f(x) = x3 - 3x2
a. Find the critical points of the following functions on the given interval.b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.c. Find the absolute maximum and minimum values on the given interval when they exist.g(x) = (x -
Determine the following indefinite integrals. dx +5/2
Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates.ln x20; ln x
Find the solution of the following initial value problems.h'(t) = 6 sin 3t; h(π/6) = 6
Locate all local maxima and minima of f(x) = x3 - 3bx2 + 3a2x + 23, where a and b are constants, in the following cases.a. |a| < |b|b. |a| > |b|c. |a| = |b|
Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.f(x) = 2x4 + 8x3 + 12x2 - x - 2
a. Find the critical points of the following functions on the given interval.b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.c. Find the absolute maximum and minimum values on the given interval when they exist.f(x) = x2/3 (4 -
Determine the following indefinite integrals. x + 1
Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates.x2 ln x; ln2 x
Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.g(t) = 3t5 - 30t4 + 80t3 + 100
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. УА y = h(x) + + х C2
Consider the function f(x) = |x - 2| + |x + 3| on [-4, 4]. Graph f, identify the critical points, and give the coordinates of the local and absolute extreme values.
Sketch a graph of the following polynomials. Identify local extrema, inflection points, and x- and y-intercepts when they exist.f(x) = x4 - 6x2
Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle’s Theorem.f(x) = 1 - x2/3; [-1, 1]
What two positive real numbers whose product is 50 have the smallest possible sum?
In terms of limits, what does it mean for the rates of growth of f and g to be comparable as x→∞?
Suppose you want to travel D miles at a constant speed of (60 + x) mi/hr, where x could be positive or negative. The time in minutes required to travel D miles is T(x) = 60D(60 + x)-1.Use the result of Exercise 9 to approximate the amount of time it takes to drive 45 miles at 62 mi/hr. What is the
For a given function f, explain the steps used to solve the initial value problem F'(t) = f(t), F(0) = 10.
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) = x3 + x2 + 1; x0 = -2
Find the values of x that minimize each function.a. f(x) = (x - 1)2 + (x - 5)2b. f(x) = (x - a)2 + (x - b)2, for constants a and bc. for a positive integer n and constants a1, a2, . . . , an. 1η fs)Σ(- a. k=1
Find the solution of the following initial value problems.g'(x) = 7x(x6 - 1/7); g(1) = 2
Suppose f is continuous on an interval containing a critical point c and f"(c) = 0. How do you determine whether f has a local extreme value at x = c?
Use analytical methods together with a graphing utility to graph the following functions on the interval [-2π, 2π]. Define f at x = 0 so that it is continuous there. Be sure to uncover all relevant features of the graph.a. b. 1 – f(x) cos x 3 .2 x cos x f(x) x2
a. Find the critical points of the following functions on the given interval.b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.c. Find the absolute maximum and minimum values on the given interval when they exist.f(θ) = 2 sin θ
Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates.x10; e0.01x
Determine the following indefinite integrals. (2x + 1)°dx
Explain how a function can have an absolute minimum value at an endpoint of an interval.
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and minimum values (if they exist). Graph the function to confirm your conclusions.g(x) = x sin-1 x on [-1, 1]
Sketch a graph of the following polynomials. Identify local extrema, inflection points, and x- and y-intercepts when they exist.f(x) = 3x - x3
Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle’s Theorem.f(x) = 1 - |x|; [-1, 1]
What two non negative real numbers a and b whose sum is 23 maximize a2 + b2? Minimize a2 + b2?
In terms of limits, what does it mean for f to grow faster than g as x→∞?
Suppose you want to travel D miles at a constant speed of (60 + x) mi/hr, where x could be positive or negative. The time in minutes required to travel D miles is T(x) = 60D(60 + x)-1.Show that the linear approximation to T at the point x = 0 is T(x) ~ L(x) = D(1 60
Two triangular pens are built against a barn. Two hundred meters of fencing are to be used for the three sides and the diagonal dividing fence (see figure). What dimensions maximize the area of the pen? Barn
If F(x) = x2 - 3x + C and F (-1) = 4, what is the value of C?
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) = x2 - 10; x0 = 4
Find the solution of the following initial value problems.g'(x) = 7x6 - 4x3 + 12; g(1) = 24
Sketch a complete graph of the following functions. Use analytical methods and a graphing utility together in a complementary way.f(x) = sin (3π cos x2 on (-π/2, π/2)
Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.h(t) = 2 + cos 2t, for -π ≤ t ≤ π
a. Find the critical points of the following functions on the given interval.b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.c. Find the absolute maximum and minimum values on the given interval when they exist.f(x) = 6x4 -
Determine the following indefinite integrals. (x8 3x³ + 1) dx
Evaluate the following limits or explain why they do not exist. Check your results by graphing. 1 1/x 2* • lim x→0+ \ 3 3* + 3 $1 1/x 2* • lim x→0+ \ 3 3* + 3$
Many species of small mammals (such as flying squirrels and marsupial gliders) have the ability to walk and glide. Recent research suggests that these animals choose the most energy-efficient means of travel. According to one empirical model, the energy required for a glider with body mass m to
Find the solution of the following initial value problems.f'(x) = 2x - 3; f(0) = 4
Sketch a complete graph of the following functions. Use analytical methods and a graphing utility together in a complementary way. xV[x? - 1| |f(x) x* + 1
Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.f(x) = √x ln x
Is it possible for a function to satisfy f(x) > 0, f'(x) > 0, and f"(x) < 0 on an interval? Explain.
Find the critical points of f. Assume a is a constant.f(x) = 1/5 x5 - a4x
Determine which of the two functions grows faster or state that they have comparable growth rates.2x and 4x/2
Evaluate the following limits or explain why they do not exist. Check your results by graphing. lim (x + cos x) х—0
A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes that intersect on a diameter D of the tree. The angle between the two half-planes is θ. Prove that for a given tree and fixed angle θ, the volume of the notch is
For the following functions f, find the antiderivative F that satisfies the given condition.f(θ) = 2 sin 2θ - 4 cos 4θ; F(π/4) = 2
How do you determine the absolute maximum and minimum values of a continuous function on a closed interval?
Sketch a complete graph of the following functions. Use analytical methods and a graphing utility together in a complementary way. sin TX on [0, 2] f(x) 1 + sin ™x
Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.f(x) = tan-1 x
Find the critical points of f. Assume a is a constant.f(x) = x3 - 3ax2 + 3a2x - a3
Determine which of the two functions grows faster or state that they have comparable growth rates.√x6 + 10 and x3
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and minimum values (if they exist). Graph the function to confirm your conclusions.f(x) = 2x ln x + 10 on (0, 4)
Evaluate the following limits or explain why they do not exist. Check your results by graphing. 10 lim
Suppose that a light source at A is in a medium in which light travels at speed v1 and the point B is in a medium in which light travels at speed v2 (see figure). Using Fermat’s Principle, which states that light travels along the path that requires the minimum travel time (Exercise 65), show
Sketch a graph of the following polynomials. Identify local extrema, inflection points, and x- and y-intercepts when they exist.f(x) = x3 - 6x2 + 9x
Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle’s Theorem.f(x) = cos 4x; [π/8, 3π/8]
For the following functions f, find the antiderivative F that satisfies the given condition. 3y + 5 f(y) F(1) = 3 y
What two non negative real numbers with a sum of 23 have the largest possible product?
Sketch a complete graph of the following functions. Use analytical methods and a graphing utility together in a complementary way.f(x) = 10x6 - 36x5 - 75x4 + 300x3 + 120x2 - 720x
Give the two-step method for attacking an indeterminate limit of the form lim f(x)* х
Use the linear approximation given in Example 1 to answer the following questions.If you travel one mile in 63 seconds, what is your approximate average speed? What is your exact speed?
Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.f(x) = e-x2/2
Evaluate ∫cos ax dx and ∫sin ax dx, where a is a constant.
Find the critical points of f. Assume a is a constant.f(x) = x√x - a
Write the formula for Newton’s method and use the given initial approximation to compute the approximations x1 and x2.f(x) = x3 - 2; x0 = 2
Determine which of the two functions grows faster or state that they have comparable growth rates.ex and 3x
Give a function that does not have an inflection point at a point where f"(x) = 0.
Sketch the graph of a function f that has a local minimum value at a point c where f'(c) is undefined.
Evaluate the following limits or explain why they do not exist. Check your results by graphing. lim (tan x)* х |х>0+
a. Two poles of heights m and n are separated by a horizontal distance d. A rope is stretched from the top of one pole to the ground and then to the top of the other pole. Show that the configuration that requires the least amount of rope occurs when θ1 = θ2 (see figure).b. Fermat’s Principle
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and minimum values (if they exist). Graph the function to confirm your conclusions.f(x) = 4x1/2 - x5/2 on [0, 4]
For the following functions f, find the antiderivative F that satisfies the given condition.f(u) = 2eu + 3; F(0) = 8
Sketch a complete graph of the following functions. Use analytical methods and a graphing utility together in a complementary way. on (1, 3) f(x) = 1 + cos X
Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.g(x) = 3√x - 4
Find the critical points of f. Assume a is a constant.f(x) = x/√x - a
Determine which of the two functions grows faster or state that they have comparable growth rates.10x and ln x2
Evaluate the following limits or explain why they do not exist. Check your results by graphing.for a constant a lim (2r + x), 1/x ах
A mathematician stands on a beach with his dog at point A. He throws a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve his ball.
Draw the graph of a function f such that f(1) = f'(1) = f"(1) = 1. Draw the linear approximation to the function at the point (1, 1). Now draw the graph of another function g such that g(1) = g'(1) = 1 and g"(1) = 10. (It is not possible to represent the second derivative exactly, but your graphs

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