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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Use the Shell Method to find the volume obtained by rotating region A in Figure 12 about the given axis.x-axis 6 2 У A 1 B y=x2+2 2 X
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.y = x2, y = 12 − x, x = 0, about y = 15
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.y = x √x − 2, y = −x √ x − 2, x = 4
Calculate the average over the given interval.ƒ(x) = x3, [−1, 1]
Find the volume obtained by rotating the region about the given axis. The regions refer to the graph of the hyperbola y2 − x2 = 1 in Figure 4.The shaded region between the upper branch of the hyperbola and the x-axis for −c ≤ x ≤ c, about the x-axis 3 2 y -2 -3+ C y=x X ye – x2 = 1
A model train of mass 0.5 kg is placed at one end of a straight 3-m electric track. Assume that a force F(x) = (3x − x2) N acts on the train at distance x along the track. Use the Work-Energy Theorem (Exercise 39) to determine the velocity of the train when it reaches the end of the track.
Use the Shell Method to find the volume obtained by rotating region A in Figure 12 about the given axis.y = −2 6 2 У A 1 B y=x2+2 2 X
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.y = 16 − 2x, y = 6, x = 0, about x-axis
Calculate the average over the given interval. f(x) = cos x, [0, 4] 6
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.x = 2y, x + 1 = (y − 1)2
With what initial velocity v0 must we fire a rocket so it attains a maximum height r above the earth? Use the results of Exercises 35 and 39. As the rocket reaches its maximum height, its KE decreases from 1/2 mv20 to zero.Data From Exercise 35The gravitational force between two objects of mass m
Find the volume obtained by rotating the region about the given axis. The regions refer to the graph of the hyperbola y2 − x2 = 1 in Figure 4.The region between the upper branch of the hyperbola and the x-axis for 0 ≤ x ≤ c, about the y-axis 3 2 y -2 -3+ C y=x X ye – x2 = 1
With what initial velocity must we fire a rocket so it attains a maximum height of r = 20 km above the surface of the earth?
Calculate the average over the given interval. f(x) = sec² x, [¹]
Find the volume of the solid obtained by rotating the region enclosed by the graphs about the given axis.y = 32 − 2x, y = 2 + 4x, x = 0, about y-axis
Use the Shell Method to find the volume obtained by rotating region A in Figure 12 about the given axis.y = 6 6 2 У A 1 B y=x2+2 2 X
Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.x + y = 1, x1/2 + y1/2 = 1
Find the volume obtained by rotating the region about the given axis. The regions refer to the graph of the hyperbola y2 − x2 = 1 in Figure 4.The region between the upper branch of the hyperbola and the line y = x for 0 ≤ x ≤ c, about the x-axis 3 2 y -2 -3+ C y=x X ye – x2 = 1
Calculate escape velocity, the minimum initial velocity of an object to ensure that it will continue traveling into space and never fall back to Earth (assuming that no force is applied after takeoff). Take the limit as r → ∞in Exercise 41.Data From Exercise 41With what initial velocity
Use the most convenient method (Disk or Shell Method) to find the volume obtained by rotating region B in Figure 12 about the given axis.y = 8 6 2 y A y=x² +2 B 2 X
Calculate the average over the given interval.ƒ(x) = 2x3 − 6x2, [−1, 3]
Evaluate the limit, using L’Hôpital’s Rule where it applies. 1²-25 lim x-5 5-4x 1²
Review the MVT stated in Section 4.3 (Theorem 1, p. 231) and show how it can be used, together with the Fundamental Theorem of Calculus, to prove the MVT for Integrals. THEOREM 1 The Mean Value Theorem Assume that f is continuous on the closed interval [a, b] and differentiable on (a, b). Then
An object is tossed into the air vertically from ground level with initial velocity v0 ft/s at time t = 0. Find the average speed of the object over the time interval [0, T], where T is the time the object returns to Earth.
Evaluate the limit, using L’Hôpital’s Rule where it applies. lim x→3 2x²5x-3 x - 4
Use a calculator to compute sinh x and cosh x for x = −3, 0, 5.
Which of the following satisfy ƒ−1(x) = ƒ(x)? (a) f(x) = x (b) f(x) = 1- x (c) f(x) = 1 (d) f(x)=√x (e) f(x) = |x| (f) f(x) = x-¹
Show that ƒ(x) = 7x − 4 is invertible and find its inverse.
Compute logb2 (b4).
Calculate without using a calculator.log327
Two quantities increase exponentially with growth constants k = 1.2 and k = 3.4, respectively.Which quantity doubles more rapidly?
What is wrong with applying L’Hôpital’s Rule to 1²-2x, lim x-0 3x-2
A certain population P of bacteria obeys the exponential growth law P(t) = 2000e1.3t (t in hours).(a) How many bacteria are present initially?(b) At what time will there be 10,000 bacteria?
Evaluate without using a calculatorcos−1 1
Compute sinh(ln 5) and tanh(3 ln 5) without using a calculator.
Which of the graphs in Figure 16 is the graph of a function satisfying ƒ−1 = ƒ? y (A) y (C) X+ X= y y (B) (D) -X -X
Compute cosh x and tanh x, assuming that sinh x = 0.8.
In 2009, 2012, and 2015, the number (in millions) of smart phones sold in the world was 172.4, 680.1, and 1423.9, respectively.(a) Let t represent time in years since 2009, and let S represent the number of smart phones sold in millions. Determine M, A, and k for a logistic model, S (t) =
Solve for x: ln(x2 + 1) − 3 ln x = ln(2).
Simplify by referring to the appropriate triangle or trigonometric identity.cot(sec−1 x)
Consider the equation Mw = 2/3 log10 E − 10.7, relating the moment magnitude of an earthquake and the energy E (in ergs) released by it.(a) Express E as a function of Mw.(b) Show that when Mw increases by 1, the energy increases by a factor of approximately 31.6.(c) Compute dE/dMw.
Refer to the appropriate triangle or trigonometric identity to compute the given value. cos (sin-¹) 3
Simplify by referring to the appropriate triangle or trigonometric identity.cot(sin−1 x)
Over rough uneven terrain, the log wind profile is expressed as With v0 = 10 m/s at h0 = 10 m, determine v and dv/dh at h = 60. In(h/0.4) In(ho/0.4) V = 107
Let n be a nonzero integer. Find a domain on which ƒ(x) = (1 − xn)1/n coincides with its inverse. The answer depends on whether n is even or odd.
Refer to the appropriate triangle or trigonometric identity to compute the given value. tan (cos-¹ داندا 3
Over open water, the log wind profile is expressed asWith v0 = 10 m/s at h0 = 10 m, determine v and dv/dh at h = 60. V = 10 In(h/0.0002) In(ho/0.0002)
Let ƒ(x) = x7 + x + 1.(a) Show that ƒ−1 exists (but do not attempt to find it). Show that ƒ is increasing.(b) What is the domain of ƒ−1?(c) Find ƒ−1(3).
If you earn an interest rate of R percent, continuously compounded, your money doubles after approximately 70/R years. For example, at R = 5%, your money doubles after 70/5 or 14 years. Use the concept of doubling time to justify the Banker’s Rule. Sometimes, the rule 72/R is used. It is less
(a) Show that at an annual rate of r, if interest is compounded n times in a year, then the amount in the account at the end of the year is A(1 + r/n)n, where A was the amount in the account at the start of the year.(b) Show that if A0 is initially invested in an account that pays annual interest
Prove that cosh−1 t = ln(t + √t2 − 1) for t > 1.
Let (a) Use the result of Exercise 68 to evaluate G(b) for all b > 0.(b) Verify your result graphically by plotting y = (1 + bx)1/x together with the horizontal line y = G(b) for the values b = 0.25, 0.5, 2, 3.Data From Exercise 68 G(b) = lim (1 + b) ¹/x.
In the following cases, check that x = c is a critical point and use Exercise 75 to determine whether ƒ(c) is a local minimum or a local maximum.Data From Exercise 75The Second Derivative Test for critical points fails if ƒ"(c) = 0. This exercise develops a Higher Derivative Test based on the
Use Exercise 116 to prove the following statements.(a) G has an inverse with domain R and range {x : x > 0}. Denote the inverse by F.(b) F(x + y) = F(x)F(y) for all x, y. It suffices to show that G(F(x)F(y)) = G(F(x + y)).(c) F(r) = Er for all numbers. In particular, F(0) = 1.(d) F(x) = F(x).
Verify that t(y) in Exercise 120 satisfies t'(y) = sec y and find a value of a such thatData From Exercise 120Show that t(y) = sinh−1(tan y) is the inverse of gd(y) for 0 ≤ y t(y) = a dt COS /
Use L’Hôpital’s Rule to prove that for all a > 0 and b > 0, lim 00+u u\u/19 + 1/11/ bling" 2 = Vab $lim 00+u u\u/19 + 1/11/ bling" 2 = Vab$
The graph of a function looks like the track of a roller coaster. Is the function one-to-one?
Use L’Hôpital’s Rule to evaluate the limit. X lim x-00 ex
Find a domain on which ƒ is one-to-one and a formula for the inverse of ƒ restricted to this domain. Sketch the graphs of ƒ and ƒ−1. f(s) = = 1 s2 S
Calculate without using a calculator. log₂ + log₂ 24
Use L’Hôpital’s Rule to evaluate the limit. lim X-→-∞0 In(x + 1)
Is it better to receive $1000 today or $1300 in 4 years? Consider r = 0.08 and r = 0.03.
Find a domain on which ƒ is one-to-one and a formula for the inverse of ƒ restricted to this domain. Sketch the graphs of ƒ and ƒ−1.ƒ(x) = √x3 + 9
Is ƒ(x) = x2 + 2 one-to-one? If not, describe a domain on which it is one-to-one.
For which values of b does ƒ(x) = bx have a negative derivative?
Give an example of an angle θ such that cos−1(cos θ) ≠ θ. Does this contradict the definition of inverse function?
When is ln x negative?
Calculate without using a calculator. log5 1 25
A cell population grows exponentially beginning with one cell. Which takes longer: increasing from one to two cells or increasing from 15 million to 20 million cells?
Find the inverse of ƒ(x) = √x3 − 8 and determine its domain and range.
A quantity P obeys the exponential growth law P(t) = e5t (t in years).(a) At what time t is P = 10?(b) At what time t is P = 20?(c) At what time t is P = 40?(d) How long does it take for P to double?
Does L’Hôpital’s Rule apply toboth approach ∞ as x → a? lim f(x)g(x) if f(x) and g(x) x→a
Evaluate the limit, using L’Hôpital’s Rule where it applies. 1³-64 lim x 4 x² + 16
For which values of x are y = sinh x and y = cosh x increasing and decreasing?
The function ƒ maps teenagers in the United States to their last names. Explain why the inverse function ƒ−1 does not exist.
What is the largest interval containing zero on which ƒ(x) = sin x is one-to-one?
For which values of b is the graph of y = bx concave up?
What is the geometric interpretation of the identityWhat does this identity tell us about the derivatives of sin−1 x and cos−1 x? sin¹x + cos¹. x = π 2
What is ln(−3)? Explain.
Calculate without using a calculator.ln 1
Find the inverse of ƒ(x) = x − 2/x − 1 and determine its domain and range.
What is wrong with saying, “To apply L’Hôpital’s Rule to the limit use the Quotient Rule to differentiate ln(1 − x)/x and then take the limit.” lim X-0 In(1-x) X
Write ƒ(t) = 5(7)t in the form ƒ(t) = P0ekt for some P0 and k.
Describe three properties of hyperbolic functions that have trigonometric analogs.
Evaluate the limit, using L’Hôpital’s Rule where it applies. 14 + 2x + 1 x1x52x1 lim
Show that y = tanh x is an odd function.
The following fragment of a train schedule for the New Jersey Transit System defines a function ƒ from towns to times. Is f one-to-one? What is ƒ−1(6:27)? Trenton Hamilton Township Princeton Junction New Brunswick 6:21 6:27 6:34 6:38
Show that ƒ(x) = x − 2/x + 3 is invertible and find its inverse.(a) What is the domain of ƒ? The range of ƒ−1?(b) What is the domain of ƒ−1? The range of ƒ?
Which point lies on the graph of y = bx for all b?
Explain the statement “The logarithm converts multiplication into addition.”
Calculate without using a calculator.log5(54)
The PV of N dollars received at time T is (choose the correct answer):(a) The value at time T of N dollars invested today(b) The amount you would have to invest today in order to receive N dollars at time T
Find a domain on which h(t) = (t − 3)2 is one-to-one and determine the inverse on this domain.
What is wrong with applying L’Hˆopital’s Rule to lim x¹/x? X→0+
Write ƒ(t) = 9e1.4t in the form ƒ(t) = P0bt for some P0 and b.
What are y(100) and y(101) for y = cosh x?
Evaluate the limit, using L’Hôpital’s Rule where it applies. 9-x+2/1¹*² lim x 9 x³/2-27
Refer to the graphs to explain why the equation sinh x = t has a unique solution for every t and why cosh x = t has two solutions for every t > 1.
A homework problem asks for a sketch of the graph of the inverse of ƒ(x) = x + cos x. Frank, after trying but failing to find a formula for ƒ−1(x), says it’s impossible to graph the inverse. Bianca hands in an accurate sketch without solving for ƒ−1. How did Bianca complete the problem?
Verify that ƒ(x) = x3 + 3 and g(x) = (x − 3)1/3 are inverses by showing that ƒ(g(x)) = x and g(ƒ(x)) = x.

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