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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

Let ƒ(t) be a function defined for all positive values of t. The Laplace Transform of ƒ(t) is defined bywhen the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises find the Laplace Transform of the function.ƒ(t) = sin at 8 · Sºr 0 F(s):
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If p(x) is a polynomial, then lim p(x) x-∞ ex 0.
Let ƒ(t) be a function defined for all positive values of t. The Laplace Transform of ƒ(t) is defined bywhen the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises find the Laplace Transform of the function.ƒ(t) = eat 8 · Sºr 0 F(s): = e-stf(t)
Let ƒ(t) be a function defined for all positive values of t. The Laplace Transform of ƒ(t) is defined bywhen the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises find the Laplace Transform of the function.ƒ(t) = t² 8 · Sºr 0 F(s): = e-stf(t)
Let ƒ(t) be a function defined for all positive values of t. The Laplace Transform of ƒ(t) is defined bywhen the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises find the Laplace Transform of the function.ƒ(t) = 1 8 · Sºr 0 F(s): = e-stf(t) dt
In Exercises 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 = et 1 then y' = 2x
Let ƒ(t) be a function defined for all positive values of t. The Laplace Transform of ƒ(t) is defined bywhen the improper integral exists. Laplace Transforms are used to solve differential equations. In Exercises find the Laplace Transform of the function.ƒ(t) = t 8 · Sºr 0 F(s): = e-stf(t) dt
The graph shows the probability density function for a car brand that has a mean fuel efficiency of 26 miles per gallon and a standard deviation of 2.4 miles per gallon.(a) Which is greater, the probability of choosing a car at random that gets between 26 and 28 miles per gallon or the probability
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. lim x->0 x + ¹] x² + x + 1 X = lim x-0 2x + 1 1 1
In Exercises apply the Extended Mean Value Theorem to the functions ƒ and g on the given interval. Find all values in the interval (a, b) such that f'(c) g'(c) f(b) f(a) - g(b) g(a)*
In Exercises apply the Extended Mean Value Theorem to the functions ƒ and g on the given interval. Find all values in the interval (a, b) such that f'(c) g'(c) f(b) f(a) - g(b) g(a)*
In Exercises apply the Extended Mean Value Theorem to the functions ƒ and g on the given interval. Find all values in the interval (a, b) such that f'(c) g'(c) f(b) f(a) - g(b) g(a)*
The average lengths (from beak to tail) of different species of warblers in the eastern United States are approximately normally distributed with a mean of 12.9 centimeters and a standard deviation of 0.95 centimeter (see figure). The probability that a randomly selected warbler has a length
In Exercises apply the Extended Mean Value Theorem to the functions ƒ and g on the given interval. Find all values in the interval (a, b) such that f'(c) g'(c) f(b) f(a) - g(b) g(a)*
The formula for the amount A in a savings account compounded n times per year for t years at an interest rate r and an initial deposit of P is given byUse L’Hôpital’s Rule to show that the limiting formula as the number of compoundings per year approaches infinity is given by A = Pert. nt A =
In Exercises determine whether the improper integral diverges or converges. Evaluate the integral if it converges. *∞o 2 √x(x + 4) dx
In Exercises determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 00 √2 1 x√√x² - 4 =dx
Find the volume of the solid generated by revolving the region bounded by the graphs of y = xe-x,y = 0, and x = 0 about the x-axis.
The board of directors of a corporation is calculating the price to pay for a business that is forecast to yield a continuous flow of profit of $500,000 per year. The money will earn a nominal rate of 5% per year compounded continuously. What is the present value of the business(a) For 20 years?(b)
The Gamma Function Γ(n) is defined in terms of the integral of the function given by ƒ(x) = xn−¹e-x, n > 0. Show that for any fixed value of n, the limit of ƒ(x) as x approaches infinity is zero.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If the graph of ƒ is symmetric with respect to the origin or they-axis, then ∫0∞ ƒ(x) dx converges if and only if ∫ ∞-∞ f(x) dx converges.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If ƒ' is continuous on [0, ∞) andthen lim f(x) = 0, x-00
The velocity of an object falling through a resisting medium such as air or water is given bywhere v0 is the initial velocity, t is the time in seconds, and k is the resistance constant of the medium. Use L'Hôpital's Rule to find the formula for the velocity of a falling body in a vacuum by fixing
In Exercises determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 1 X dx
In Exercises determine whether the improper integral diverges or converges. Evaluate the integral if it converges. ∞ In x x² dx
In Exercises graph ƒ(x)/g(x)and ƒ'(x)/g '(x) near x = 0. What do you notice about theseratios as x → 0? How does this illustrate L'Hôpital's Rule? f(x) = sin 3x, g(x) = sin 4x
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ is continuous on [0, ∞) and ∫0∞ ƒ(x) dx diverges, then lim f(x) = 0. x→∞
In Exercises graph ƒ(x)/g(x) and ƒ'(x)/g '(x) near x = 0. What do you notice about these ratios as x → 0? How does this illustrate L'Hôpital's Rule? f(x) = ³x - 1, g(x) = = X
In Exercises determine whether the improper integral diverges or converges. Evaluate the integral if it converges. e-1/x x² dx
A "semi-infinite" uniform rod occupies the nonnegative x-axis. The rod has a linear density 8, which means that a segment of length dx has a mass of 8 dx. A particle of mass M is located at the point (-a, 0). The gravitational force F that the rod exerts on the mass is given bywhere G is the
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If ƒ is continuous on [0, ∞) and converges. lim f(x) = 0, then fo f(x) dx x →∞
In Exercises determine whether the improper integral diverges or converges. Evaluate the integral if it converges. xp x u[ zx 00.
The magnetic potential P at a point on the axis of a circular coil is given bywhere N, I, r, k, and c are constants. Find P. P= [" (1². 2π NIr k 1 (p² + x²) ³/2 dx
In Exercises (a) Explain why L’Hôpital’s Rule cannot be used to find the limit (b) Find the limit analytically(c) Use a graphing utility to graph the function and approximate the limit from the graph. Compare the result with that in part (b). lim X-∞ X √x² + 1
In Exercises (a) Explain why L’Hôpital’s Rule cannot be used to find the limit (b) Find the limit analytically(c) Use a graphing utility to graph the function and approximate the limit from the graph. Compare the result with that in part (b). lim x→π/2 tan x sec X
In Exercises determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 7 6² x ²1 2² X dx
In Exercises find the capitalized cost C of an asset (a) For n = 5 years(b) For n = 10 years(c) Forever. The capitalized cost is given bywhere C0 is the original investment, t is the time in years, r is the annual interest rate compounded continuously, and c(t) is the annual cost of
In Exercises determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 16 [1²0 / 1² dx
L’Hôpital’s Rule is used incorrectly. Describe the error. lim X-∞ e-x -e-x = lim xe-x lim 1 x- X-8 = 1
L’Hôpital’s Rule is used incorrectly. Describe the error. lim xcos 818 1 lim X10 Tim = 0 cos(1/x) 1/x - sin(1/x)](1/x²) -1/x²
In Exercises use L’Hôpital’s Rule to evaluate the limit. lim x→1+ 2 ln x In 2 x - 1
L’Hôpital’s Rule is used incorrectly. Describe the error. lim x→0 e2x - 1 et = lim 40 2e²f ex lim 2ex x=0 = 2
In Exercises find the capitalized cost C of an asset (a) For n = 5 years(b) For n = 10 years(c) Forever. The capitalized cost is given bywhere C0 is the original investment, t is the time in years, r isthe annual interest rate compounded continuously, and c(t) isthe annual cost of
In Exercises use L’Hôpital’s Rule to evaluate the limit. lim 1000 1 + n→∞0 0.09 n n
A nonnegative function ƒ is called a probability density function ifThe probability that x lies between a and b is given by The expected value of x is given byIn Exercises (a) Show that the nonnegative function is a probability density function(b) Find P(0 ≤ x ≤ 4)(c) Find E(x). f(t)
In Exercises use L’Hôpital’s Rule to evaluate the limit. lim (x - 1)In x x-1+
L’Hôpital’s Rule is used incorrectly. Describe the error. 3x² + 4x + 1 x=2x²²x2 lim tim x-2 6x +4 2x - T lim x-2-2 3
A nonnegative function ƒ is called a probabilitydensity function ifThe probability that x lies between a and b is given byThe expected value of x is given byIn Exercises (a) Show that the nonnegative functionis a probability density function(b) Find P(0 ≤ x ≤ 4)(c) Find E(x). f(t) dt = 1.
In Exercises use L’Hôpital’s Rule to evaluate the limit. lim (In x)²/x x →∞
In Exercises find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function. y In x X
In Exercises use L’Hôpital’s Rule to evaluate the limit. lim xe-x x10
In Exercises use L’Hôpital’s Rule to evaluate the limit. e2r lim xxx²
In Exercises use the weight of the rocket to answer each question. (Use 4000 miles as the radius of Earth and do not consider the effect of air resistance.)(a) How much work is required to propel the rocket an unlimited distance away from Earth’s surface?(b) How far has the rocket traveled when
In Exercises find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function.y = 2xe-x
In Exercises use the weight of the rocket to answer each question. (Use 4000 miles as the radius of Earth and do not consider the effect of air resistance.)(a) How much work is required to propel the rocket an unlimited distance away from Earth’s surface?(b) How far has the rocket traveled when
In Exercises find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function.y = xx, x > 0
Find the area of the surface formed by revolving the graph of y = 2e-x on the interval [0, ∞) about the x-axis.
In Exercises find any asymptotes and relative extrema that may exist and use a graphing utility to graph the function.y = x¹/x, x > 0
The region bounded by (x - 2)2 + y² = 1 is revolved about the y-axis to form a torus. Find the surface area of the torus.
In Exercises find the area of the unbounded shaded region. y = - ln x y 3 2- 1 2 3 4
In Exercises find the area of the unbounded shaded region. y = et, -∞0 < x ≤ 1 -3 -2 - 3 y 2+ -¹4 X
Complete the table to show that x eventually “overpowers” (In x)4. X (In x)4 X 10 10² 104 106 108 10¹⁰
In Exercises find the area of the region. y 1 0.5 1 25x² y 2 4
In Exercises find the area of the region. y=x√√4 - x 4 3 2 y 1 2 3 نیا 4 X
Consider the integralTo determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals if the given integral converges? 3 10 x² - 2x - dx.
Use the graph of ƒ to find the limit.(a)(b)(c) 6 4 2 2 f(x)= 3 In x 4 6 8 4 x-1 ➤X
Determine which of the following limits can be evaluated using L’Hôpital’s Rule. Explain your reasoning. Do not evaluate the limit.(a)(b)(c)(d)(e)(f) lim x - 2 9- x - x x 6
In Exercises evaluate the definite integral using any method. Use a graphing utility to verify your result. So X √4 + x dx
In Exercises evaluate the definite integral using any method. Use a graphing utility to verify your result. TT x sin x dx
Find differentiable functions ƒ and g such thatExplain how you obtained your answers. lim f(x) = lim g(x) ) = ∞ and lim [f(x) − g(x)] = 25. -
Define the terms converges and diverges when working with improper integrals.
In Exercises evaluate the definite integral using any method. Use a graphing utility to verify your result. So xe3x dx
Describe the different types of improper integrals.
Find differentiable functions ƒ and g that satisfy the specified condition such thatExplain how you obtained your answers.(a)(b)(c) lim f(x) = 0 0 X-5 and and lim g(x) = 0. x-5
In Exercises use to determine whether the improper integral converges or diverges. 1 et + x dx
In Exercises evaluate the definite integral using any method. Use a graphing utility to verify your result. In x X dx
In Exercises evaluate the definite integral using any method. Use a graphing utility to verify your result. X (x − 2)(x − 4) dx
In Exercises use to determine whether the improper integral converges or diverges. 1 - sin x x² -dx
In Exercises evaluate the definite integral using any method. Use a graphing utility to verify your result. √√5 J2 x(x² - 4)³/2 dx
List six different indeterminate forms.
In Exercises use to determine whether the improper integral converges or diverges. ∞o 1 √x(x + 1) dx
In Exercises (a) Describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim 10 3 x² X+0+x
In Exercises solve the differential equation using any method. y' =√1 cos 0 1-
In Exercises use to determine whether the improper integral converges or diverges. roo J2 1 3/x(x - 1) dx
In Exercises (a) Describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim x-1+ 3 2 ln x In x - 1)
In Exercises use to determine whether the improper integral converges or diverges. *00 J2 1 /x-1 dx
In Exercises solve the differential equation using any method. y' = ln(x² + x)
In Exercises (a) Describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). 1 x² -
In Exercises solve the differential equation using any method. dy √√√4x² dx 2x
In Exercises (a) Describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). 8 2-4 lim x2+x² 1 X x - 2)
In Exercises use to determine whether the improper integral converges or diverges. roo 1 x² + 5 dx
In Exercises solve the differential equation using any method. dy dx || 25 x² - 25
In Exercises use to determine whether the improper integral converges or diverges. x¹ ex dx
In Exercises find the indefinite integral using any method. (sin + cos 0)² de 0
In Exercises (a) Describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim cos X-0+ 2 X
In Exercises use to determine whether the improper integral converges or diverges. 1 X dx
In Exercises (a) Describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim (In x)x-1 x-1+
In Exercises find the indefinite integral using any method. S cos x In(sin x) dx
In Exercises (a) Describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim [3(x-4)]*-4 X-4+
In Exercises find the indefinite integral using any method. 3x³ + 4x (x² + 1)2 dx

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