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mathematics
calculus 10th edition

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

In Exercises determine the convergence or divergence of the series. n=1 1 + k-n \n
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 a ≤ b + c, and a diverges, then the series b, and n=1 ∞ n=1 ∞ Cn both diverge. n=1
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.) 1 2 3 . 2 3 4 . 3 4 4 5' 5.6'
In Exercises find the positive values of p for which the series converges. 1 n3 n ln n[In(In n)]P n=
In Exercises find the positive values of p for which the series converges. n=1 n
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 an + b ≤ c and n and n=1 n=1 C₁ converges, then the series b, both converge. n n=1 an
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.) 2,1 + 1 + 1 + 1 + ½¾, . . .
In Exercises determine the convergence or divergence of the series. 00 n=1 1 In- n
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 0 < an+10 ≤ b, and n=1 b, converges, then n=1 an converges.
In Exercises find the positive values of p for which the series converges. Σ n(1 + n?)P n=1
In Exercises determine the convergence or divergence of the series. n=2 n In n
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 0 < a ≤ b, and n=1 an converges, then n=1 bn b, diverges.
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.)2, 24, 720, 40,320, 3,628,800, . . .
In Exercises find the positive values of p for which the series converges. n=1 n (1 + n²) P
In Exercises determine the convergence or divergence of the series. 18 n=0 3 5n
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.) دانی نات +15
In Exercises find the positive values of p for which the series converges. In n пр n=2
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.) 11 1,-,-...
In Exercises determine the convergence or divergence of the series. n=1 1 n+ 1 1 n+2
In Exercises determine the convergence or divergence of the series. 18 n= n 3 ท
It appears that the terms of the seriesare less than the corresponding terms of the convergent seriesIf the statement above is correct, then the first series converges. Is this correct? Why or why not? Make a statement about how the divergence or convergence of a series is affected by the inclusion
In Exercises find the positive values of p for which the series converges. n=2 1 n(In n)P
The graphs show the sequences of partial sums of the p-seriesUsing Theorem 9.11, the first series diverges and the second series converges. Explain how the graphs show this.Data from in Theorem 9.11 n=1 1 nº.4 and n=1 1 1.5*
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.)-2, 1, 6, 13, 22, . . .
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.) 1 1 1 1, 2, 6, 24, 120,
In Exercises determine the convergence or divergence of the series. ÿ n=1 n 1 n+ 2
In Exercises determine the convergence or divergence of the series. n=1 4n + 1 3n - 1
Use a graph to show that the inequality is true. What can you conclude about the convergence or divergence of the series? Explain.(a)(b) 1 n 00 1 法 X =dx
Let ƒ be a positive, continuous, anddecreasing function for x ≥ 1, such that an = ƒ(n). Use a graph to rank the following quantities in decreasing order. Explain your reasoning.(a)(b)(c) 7 n=2 an
In Exercises determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. an = COS TTN n²
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.)2, 8, 14, 20, . . .
A friend in your calculus class tells you that the following series converges because the terms are very small and approach 0 rapidly. Is your friend correct? Explain. 1 10,000 + 1 10,001 + 1 10,002 +
In Exercises determine the convergence or divergence of the series. n=1 n + 10 10n + 1
In Exercises determine the convergence or divergence of the sequence with the given nth term. If the sequence converges, find its limit. an || sin n n
Explain why careful analysis is required to determine the convergence or divergence of a series and why only considering the magnitudes of the terms of a series could be misleading.
(a) Evaluate the integrals(b) Use Wallis’s Formulas to prove thatfor all positive integers n. La (1-x²) dx and Lo (1-x²)² dx.
The line x = 1 is tangent to the unit circle at A. The length of segment QA equals the length of the circular arc (see figure). Show that the length of segment OR approaches 2 as P approaches A. PA
Find the value of the positive constant c such that x+c\x lim xxx C = 9.
Find the value of the positive constant c such that x - lim xx+c C X || 1 4*
Find the centroid of the region above the x-axis and bounded above by the curve y = e-c²x², where c is apositive constant (see figure). fe Show that e-c²x² dx y = 1 500 e-x² dx. y = e-c²x² X
The segment BD is the height of ΔOAB. Let R be the ratio of the area of ΔDAB to that of the shaded region formed by deleting ΔOAB from the circular sector subtended by angle 0 (see figure). Find lim R. 0→0+
Use a graphing utility to estimate each limit. Then calculate each limit using L'Hôpital's Rule. What can you conclude about the form 0 . ∞?(a)(b)(c) lim cot x + x-0+ X
Consider the problem of finding the area of the region bounded by the x-axis, the line x = 4, and the curve(a) Use a graphing utility to graph the region and approximate its area.(b) Use an appropriate trigonometric substitution to find the exact area.(c) Use the substitution x = 3 sinh u to find
Area Factor the polynomial p(x) = x4 + 1 and then find the area under the graph of y 1 x4 + 1' 0≤x≤ 1 (see figure).
Find the arc length of the graph of the function y = ln(1 − x²) on the interval 0 ≤ x ≤ 1/2 (see figure). X ·la Je
In Exercises use integration by parts to find the indefinite integral. x² sin 2x dx
Some elementary functions, such as ƒ(x) = sin(x²), do not have antiderivatives that are elementary functions. Joseph Liouville proved thatdoes not have an elementary antiderivative. Use this fact to prove thatis not elementary. ex - dx X
Suppose the denominator of a rational function can be factored into distinct linear factorsfor a positive integer n and distinct real numbers C₁, C₂,..., Cn. If N is a polynomial of degree less than n, show thatwhere Pk = N(Ck)/D'(ck) for k = 1, 2,..., n. Note that thisis the partial fraction
The velocity v (in feet per second) of a rocket whose initial mass (including fuel) is m is given bywhere u is the expulsion speed of the fuel, r is the rate at which the fuel is consumed, and g = -32 feet per second per second is the acceleration due to gravity. Find the position equation for a
Use the result of Exercise 14 to find the partial fraction decomposition ofData from in Exercise 14Suppose the denominator of a rational function can be factored into distinct linear factorsfor a positive integer n and distinct real numbers C₁, C₂,..., Cn. If N is a polynomial of degree less
Suppose that ƒ(a) = ƒ(b) = g(a) = g(b) = 0 and the second derivatives of ƒ and g are continuous on the closed interval [a, b]. Prove that b b [*f(x) g(x) dx = [*1*(x)g(x) dx. f(x)g"(x) a a
(a) Use the substitution = π/2 - x to evaluate the integral(b) Let n be a positive integer. Evaluate the integral π/2 0 sin x cos x + sin x dx.
Suppose that ƒ(a) = ƒ(b) = 0 and the second derivatives of ƒ exist on the closed interval [a, b]. Prove that "b So = 25²56² (x - a)(x - b)f"(x) dx = 2 2 f(x) dx.
Using the inequalityfor x ≥ 2 approximate + 10 x15 1 x³ - 1 x10 + 2 x15 20
Consider the shaded region between the graph of y = sin x, where 0 ≤ x ≤ π, and the line y = c, where0 ≤ c ≤ 1 (see figure). A solid is formed by revolving theregion about the line y = c.(a) For what value of does the solid have minimum volume?(b) For what value of does the solid have
In Exercises determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 8 J4 x² - 16 x² dx
Complete the table to show that ex eventually “overpowers” x5. X 15 10 20 30 40 50 100
In Exercises use L'Hôpital's Rule to determine the comparative rates of increase of the functions ƒ(x) = xm, g(x) = enx, and h(x) = (In x)n, where n > 0, m > 0, and x → ∞. x³ lim x→∞0 e²x
For what value of c does the integral converge? Evaluate the integral for this value of c. ∞ SG CX x2 + 2 3x dx
In Exercises use L’Hôpital’s Rule to evaluate the limit. (In x)² lim x1 x 1
In Exercises use L’Hôpital’s Rule to evaluate the limit. sin TX lim x-0 sin 5 mx
In Exercises use L'Hôpital's Rule to determine the comparative rates of increase of the functions ƒ(x) = xm, g(x) = enx, and h(x) = (In x)n, where n > 0, m > 0, and x → ∞. lim 01x (In x)" xm
In Exercises use L'Hôpital's Rule to determine the comparative rates of increase of the functions ƒ(x) = xm, g(x) = enx, and h(x) = (In x)n, where n > 0, m > 0, and x → ∞. Xm lim x→co ex
In Exercises approximate to two decimal places the arc length of the curve over the given interval. Function y = sin² x Interval [0, a]
In Exercises use L'Hôpital's Rule to determine the comparative rates of increase of the functions ƒ(x) = xm, g(x) = enx, and h(x) = (In x)n, where n > 0, m > 0, and x → ∞. lim X-∞ (In x)² x³
In Exercises consider the region satisfying the inequalities. (a) Find the area of the region. (b) Find the volume of the solid generated by revolving the region about the x-axis. (c) Find the volume of the solid generated by revolving the region about the y-axis. y ≤ ex, y ≥ 0,
In Exercises approximate to two decimal places the arc length of the curve over the given interval. Function y = sin x Interval [0, π]
Find the arc length of the graph of y = √16 - x² over the interval [0, 4].
In Exercises consider the region satisfying the inequalities. (a) Find the area of the region. (b) Find the volume of the solid generated by revolving the region about the x-axis. (c) Find the volume of the solid generated by revolving the region about the y-axis. 1 y ≤ s , y ≥
In Exercises use L'Hôpital's Rule to determine the comparative rates of increase of the functions ƒ(x) = xm, g(x) = enx, and h(x) = (In x)n, where n > 0, m > 0, and x → ∞. lim x →∞0 (In x)³ X
In Exercises find the area of the unbounded shaded region. Witch of Agnesi: 8 x² + 4 y -6 -4 -2 6 4 y -2 -4 -6 + 2 4 6 X
Sketch the graph of the hypocycloid of four cusps x2/3 + y2/3 = 4 and find its perimeter.
In Exercises find the area of the unbounded shaded region. Witch of Agnesi: 1 x² + 1 y -3 -2 -1 y 3- 2 -2 -3+ 1 2 3 X
In Exercises find the centroid of the region bounded by the graphs of the equations. (x - 1)² + y² = 1, (x-4)² + y² = 4
EvaluateIn L’Hôpital’s 1696 calculus textbook, he illustrated his rule using the limit of the function Tim 24-11 at x-00 a 11/x where a > 0, a ‡ 1.
In Exercises find the centroid of the region bounded by the graphs of the equations. y = √√√1-x², y = 0
Show that the indeterminate forms 00, ∞0, and 1 do not always have a value of 1 byevaluating each limit.(a)(b)(c) lim xln 2/(1+In x) X-0+
In Exercises use L'Hôpital's Rule to determine the comparative rates of increase of the functions ƒ(x) = xm, g(x) = enx, and h(x) = (In x)n, where n > 0, m > 0, and x → ∞. lim x→∞o esx X 5x
Letbe convergent and let a and b be real numbers where a ≠ b Show that ∞ f(x) dx ∞o
In L’Hôpital’s 1696 calculus textbook, he illustrated his rule using the limit of the functionas x approaches a, a > 0. Find this limit. f(x) = = /2a³x - x² = a√/a²x aax³
Prove the following generalization of the Mean Value Theorem. If ƒ is twice differentiable on the closed interval [a, b], then b ƒ(b) — ƒ(a) = f'(a)(b − a) – [*ƒ″(1)(1 - - - - a f"(t)(t - b) dt.
In Exercises rewrite the improper integral as a proper integral using the given u-substitution. Then use the Trapezoidal Rule with n = 5 to approximate the integral. So COS X √1-x =dx, u = u = √√√1-x
Prove that if ƒ(x) ≥ 0,and lim f(x) = 0, x-a
Prove that if ƒ(x) ≥ 0,and lim f(x) = 0, x→a
In Exercises rewrite the improper integral as a proper integral using the given u-substitution. Then use the Trapezoidal Rule with n = 5 to approximate the integral. S' sin x X dx, =√x U =
Find the volume of the solid generated by revolving the region bounded by the graph of about the X-axis. f(x) [x In x, 0, 0 < x≤ 2 x = 0
For what value of c does the integralconverge? Evaluate the integral for this value of c. ∞o . ( C ਜਜ +1 1 x² + 1 dx
Consider the limit(a) Describe the type of indeterminate form that is obtained by direct substitution.(b) Evaluate the limit. Use a graphing utility to verify the result. lim (-x ln x). x-0+
Use a graphing utility to graphfor k= 1, 0.1, and 0.01. Then evaluate the limit f(x) = xk - 1 k
Let ƒ''(x) be continuous. Show that lim h→0 f(x + h) 2f(x) + f(x - h) h² - = f'(x).
Find the values of a and b such that lim x-0 a - cos bx x² 2.
(a) Let ƒ'(x) be continuous. Show that(b) Explain the result of part (a) graphically. lim h→0 f(x+h)-f(x - h) 2h h) = f'(x).
Find the volume of the solid generated by revolving the unbounded region lying between y = - ln x and the y-axis (y ≥ 0) about the x-axis.
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) = sinh at 8 · Sºr 0 F(s):
In Exercises find the value of that makes the function continuous at x = 0. f(x)= = [(ex + x) ¹/x, lc, x # 0 x=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)= cosh at 8 · Sºr 0 F(s):
In Exercises find the value of that makes the function continuous at x = 0. f(x) = 4x - 2 sin 2x 2x³ x = 0 x = 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) = cos at 8 · Sºr 0 F(s):
Find the limit, as approaches 0, of the ratio of the area of the triangle to the total shaded area in the figure. (-x, 1 - cos x) -T RIN 2 2 1 y f(x) = 1- cos x RIN (x, 1 - cos x) + π X
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 lim 818 f(x) g(x) = 1, then lim [f(x) = g(x)] = 0. - 818

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