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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.a(t) = 0.2 t; v(0) = 0, s(0) = 1
Determine the following indefinite integrals. Check your work by differentiation. бх? бr2 dx * 4x4 х
Use a graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, asymptotes, intervals where the function is increasing/decreasing, and intervals of concavity.f(x) = x/ln x
Find the intervals on which f is increasing and decreasing.f(x) = -2x4 + x2 + 10
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) = x2√x + 1 on [-1, 1]
Sketch the graph of a function that is continuous on (-∞, ∞) and satisfies the following sets of conditions.f"(x) > 0 on (- ∞, -2); f"(-2) = 0; f'(-1) = f'(1) = 0; f"(2) = 0; f'(3) = 0; f"(x) > 0 on (4, ∞)
Prove thatwhere a, b, and c are positive real numbers. a" + b" + c'\l/r lim 1/r Vabc. 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.p(x) = xe-x2
Use analytical methods to evaluate the following limits. lim (x²e/* – x² – x) x)
The intensity of a light source at a distance is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. Two light sources, one twice as strong as the other, are 12 m apart. At what point on the line segment joining the sources
Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.a(t) = 4; v(0) = -3, s(0) = 2
Explain why if a runner completes a 6.2-mi (10-km) race in 32 min, then he must have been running at exactly 11 mi/hr at least twice in the race. Assume the runner’s speed at the finish line is zero.
Use Newton’s method to approximate the roots of f(x) = 3x3 - 4x2 + 1 to six digits.
Use a graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, asymptotes, intervals where the function is increasing/decreasing, and intervals of concavity. x sin x on [-27,
Evaluate the following limits. y? + y – 6 lim /8 – y? y-2 V8 – y² – y
Bamboo belongs to the grass family and is one of the fastest-growing plants in the world.a. A bamboo shoot was 500 cm tall at 10.00 a.m. and 515 cm at 3:00 p.m. Compute the average growth rate of the bamboo shoot in cm/hr over the period of time from 10:00 a.m. to 3:00 p.m.b. Based on the Mean
Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.f(x) = x/6 - sec x on [0, 8]
Carry out the following steps for the given functions f and points a.a. Find the linear approximation L to the function f at the point a.b. Graph f and L on the same set of axes.c. Based on the graphs in part (a), state whether linear approximations to f near a are underestimates or
Determine the following indefinite integrals. Check your work by differentiation.∫( 5√r2 dr
Find the intervals on which f is increasing and decreasing.f(x) = x2 - 2 ln x
Sketch the graph of a function f continuous on [a, b] such that f, f', and f" have the signs indicated in the following table on [a, b]. There are eight different cases lettered A–H and eight different graphs.
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) = x2 - 2 ln (x2 + 1)
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.g(x) = e-x2/2
A load must be suspended 6 m below a high ceiling using cables attached to two supports that are 2 m apart (see figure). How far below the ceiling (x in the figure) should the cables be joined to minimize the total length of cable used? 2 m х 6 m
Compare carefully to Exercise 33. A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to another officer 30 mi along the highway. When the car reached the location of the second officer 30 min later, it was clocked going 60 mi/hr. Can the patrol officer conclude
Evaluate the following limits. v - 1 - Vv² – 5 lim v→3
The population of a culture of cells grows according to the function where t ≥ 0 is measured in weeks.a. What is the average rate of change in the population over the interval [0, 8]?b. At what point of the interval [0, 8] is the instantaneous rate of change equal to the average rate of change?
Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.f(x) = cos 2x - x2 + 2x
Evaluate and confirm your result by graphing. х lim x→0+ and lim x→0* 1 – e*
Carry out the following steps for the given functions f and points a.a. Find the linear approximation L to the function f at the point a.b. Graph f and L on the same set of axes.c. Based on the graphs in part (a), state whether linear approximations to f near a are underestimates or
Use analytical methods to evaluate the following limits. 1 lim x3 sin х
Determine the following indefinite integrals. Check your work by differentiation. 3 3 dx х*
Find the intervals on which f is increasing and decreasing.f(x) = -12x5 + 75x4 - 80x3
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) = 1/x + ln x
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 ln x
A grain silo consists of a cylindrical concrete tower surmounted by a metal hemispherical dome. The metal in the dome costs 1.5 times as much as the concrete (per unit of surface area). If the volume of the silo is 750 m3, what are the dimensions of the silo (radius and height of the cylindrical
A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to a patrol officer 30 mi along the highway. When the car reached the location of the second officer 28 min later, it was clocked going 60 mi/hr. The driver of the car was given a ticket for exceeding the
Evaluate the following limits.n is a positive integer. x" – 1 lim х—1 х — 1 "
The energy E (in joules) released by an earthquake of magnitude M is modeled by the equation E(M) = 25,000 • 101.5 M. Approximate the change in energy released when the magnitude changes from 7.0 to 7.5 (recall that ΔE ≈ E'(a)ΔM).
Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.f(x) = cos x - x/7
Carry out the following steps for the given functions f and points a.a. Find the linear approximation L to the function f at the point a.b. Graph f and L on the same set of axes.c. Based on the graphs in part (a), state whether linear approximations to f near a are underestimates or
Determine the following indefinite integrals. Check your work by differentiation.∫(4z1/3 - z-1/3) dz
Find the intervals on which f is increasing and decreasing.f(x) = ln |x|
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) = sin x cos x on [0, 2π]
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 x)/x2
Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.a(t) = -32; v(0) = 20, s(0) = 0
a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.b. Suppose that in part (a) the
Use a graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, asymptotes, intervals where the function is increasing/decreasing, and intervals of concavity. V4x2 + 1 x2 + 1
Avalanche forecasters measure the temperature gradient dT/dh, which is the rate at which the temperature in a snowpack T changes with respect to its depth h. A large temperature gradient may lead to a weak layer in the snowpack. When these weak layers collapse, avalanches occur. Avalanche
Evaluate the following limits. x3 – x2 – 5x – 3 5х — 3 lim x-1 x4 + 2x3 - x? - 4x – - x² – 4x – 2
The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t2, where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h'(a)Δt).
An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions.
Carry out the following steps for the given functions f and points a.a. Find the linear approximation L to the function f at the point a.b. Graph f and L on the same set of axes.c. Based on the graphs in part (a), state whether linear approximations to f near a are underestimates or
Determine the following indefinite integrals. Check your work by differentiation.∫(3x + 1(4 - x) dx
Find the intervals on which f is increasing and decreasing.f(x) = tan-1 x
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) = (ex + e-x)/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. Зт Зп |f(x) = x + tan x on
Several mathematical stories originated with the second wedding of the mathematician and astronomer Johannes Kepler. Here is one: While shopping for wine for his wedding, Kepler noticed that the price of a barrel of wine (here assumed to be a cylinder) was determined solely by the length d of a
The figure shows the graphs of f, f', and f". Which curve is which? УА A х
Find all points on the interval (1, 3) at which the slope of the tangent line equals the average rate of change of f on [1, 3]. Reconcile your results with the Mean Value Theorem. yA y = f(x) + + 1 4- 3. 3.
Evaluate the following limits. tanx — п/2 -1 lim 1/х х—0о
Use linear approximation to estimate the following quantities. Choose a value of a to produce a small error.tan-1 1.05
Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error.cos 31°
Determine the following indefinite integrals. Check your work by differentiation.∫(6 3√x dx
Find the intervals on which f is increasing and decreasing.f(x) = x2√9 - x2 on (-3, 3)
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) = 12x5 - 20x3 on [-2, 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.g(x) = x2 ln x
Compare the growth rates of ln x, ln (ln x), and ln (ln (ln x)).
a. 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 line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?b. Is it possible to construct a rectangle with a greater
Find all points on the interval (1, 3) at which the slope of the tangent line equals the average rate of change of f on [1, 3]. Reconcile your results with the Mean Value Theorem. 3. 2. y = f(x) + + х 1 3 4
Use analytical methods to evaluate the following limits. lim (T – 2x) tan x x→#/2
Evaluate the following limits. 1/x lim 1/x
Use linear approximation to estimate the following quantities. Choose a value of a to produce a small error.1/4.22
An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions.
Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error.1/3√510
Determine the following indefinite integrals. Check your work by differentiation.∫(3x1/3 + 4x-1/3 + 6) dx
Find the intervals on which f is increasing and decreasing.f(x) = x2/3 (x2 - 4)
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) = x/(x2 + 1)
Given the following velocity functions of an object moving along a line, find the position function with the given initial position. Then graph both the velocity and position functions.v(t) = 2 sin 2t; s(0) = 0
Use a graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, asymptotes, intervals where the function is increasing/decreasing, and intervals of concavity. -1 tanx f(x) : .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.g(t) = e-t sin t on [-π, π]
A rectangular flower garden with an area of 30 m2 is surrounded by a grass border 1 m wide on two sides and 2 m wide on the other two sides (see figure). What dimensions of the garden minimize the combined area of the garden and borders? 2 m Flower garden 1 m
By visual inspection, locate all points on the interval [-4, 4] at which the slope of the tangent line equals the average rate of change of the function on the interval [-4, 4]. УА y х 4 -2
Evaluate the following limits. sin x – x lim х- 7x3 Zr3
a. Find the linear approximation to f at the given point a.b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?f(x) = sin-1 x; a = 1/2; f(0.48)
An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions.
Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error.1/√119
Determine the following indefinite integrals. Check your work by differentiation.∫(5m (12m3 - 10m) dm
Find the intervals on which f is increasing and decreasing.f(x) = cos2 x on [-π, π]
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) = 4x5/5 - 3x3 + 5 on [-2, 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√x + 3
The following figures show the graphs of three functions (graphs a–c). Match each function with its first derivative (graphs d–f) and its second derivative (graphs g–i). (b) УД (c) (d) У (e) (f) У. (8) (h) У (i)
Imagine a flat-bottomed cylindrical pot with a circular cross section of radius 4. A marble with radius 0 < r < 4 is placed in the bottom of the pot. What is the radius of the marble that requires the most water to cover it completely?
Find all functions f whose derivative is f'(x) = x + 1.
Evaluate the following limits. e* – sin x – 1 lim x→0 x4 + 8x³ + 12x2
a. Find the linear approximation to f at the given point a.b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?f(x) = x2/3; a = 27; f(29)
Determine whether the following statements are true and give an explanation or counterexample. a. Newton’s method is an example of a numerical method for approximating the roots of a function.b. Newton’s method gives a better approximation to the roots of a quadratic equation than the
Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error.e0.06
A rocket is launched vertically upward with an initial velocity of 120 m/s from a platform that is 125 m above the ground. Assume that the only force at work is gravity. Determine and graph the velocity and position functions of the rocket, for t ≥ 0. Then describe the motion in words.
Use analytical methods to evaluate the following limits. lim (Vx – 2 – Vx – 4)
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