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mathematics
calculus early transcendentals

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

Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.sin 0.3, n = 4
Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.f(x) = 1/(1 - x), a = 0
Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.f(x) = sin x, a = π/2
Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.f(x) = cos x, a = π/2
Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.f(x) = e-x, a = 0
Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.f(x) = cos 2x, a = 0
Find the remainder Rn for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.f(x) = sin x, a = 0
a. Approximate the given quantities using Taylor polynomials with n = 3.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.tanh 0.5
a. Approximate the given quantities using Taylor polynomials with n = 3.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.sinh 0.5
a. Approximate the given quantities using Taylor polynomials with n = 3.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.3√126
a. Approximate the given quantities using Taylor polynomials with n = 3.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.√101
a. Approximate the given quantities using Taylor polynomials with n = 3.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.4√79
a. Approximate the given quantities using Taylor polynomials with n = 3.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.√1.06
a. Approximate the given quantities using Taylor polynomials with n = 3.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.ln 1.05
a. Approximate the given quantities using Taylor polynomials with n = 3.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.tan (-0.1)
a. Approximate the given quantities using Taylor polynomials with n = 3.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.cos (-0.2)
a. Approximate the given quantities using Taylor polynomials with n = 3.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.e0.12
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = ex, a = ln 2
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = tan-1 x + x2 + 1, a = 1
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = 4√x, a = 16
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = ln x, a = e
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = 3√x, a = 8
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = √x, a = 9
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = cos x, a = π/6
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = sin x, a = π/4
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = 8√x, a = 1
a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = x3, a = 1
a. Use the given Taylor polynomial p2 to approximate the given quantity.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.Approximate 1/1.123 using f(x) = 1/(1 + x)3 and p2(x) = 1 - 3x + 6x2.
a. Use the given Taylor polynomial p2 to approximate the given quantity.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.Approximate e-0.15 using f(x) = e-x and p2(x) = 1 - x + x2/2.
a. Use the given Taylor polynomial p2 to approximate the given quantity.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.Approximate ln 1.06 using f(x) = ln (1 + x) and p2(x) = x - x2/2.
a. Use the given Taylor polynomial p2 to approximate the given quantity.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.Approximate 1/√1.08 using f(x) = 1/√1 + x and p2(x) = 1 - x/2 + 3x2/8.
a. Use the given Taylor polynomial p2 to approximate the given quantity.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.Approximate 3√1.1 using f(x) = 3√1 + x and p2(x) = 1 + x/3 - x2/9.
a. Use the given Taylor polynomial p2 to approximate the given quantity.b. Compute the absolute error in the approximation assuming the exact value is given by a calculator.Approximate √1.05 using f(x) = √1 + x and p2(x) = 1 + x/2 - x2/8.
a. Find the nth-order Taylor polynomials of the given function centered at 0, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = sin-1x
a. Find the nth-order Taylor polynomials of the given function centered at 0, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = (1 + x)-3
a. Find the nth-order Taylor polynomials of the given function centered at 0, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = (1 + x)-2
a. Find the nth-order Taylor polynomials of the given function centered at 0, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = tan x
a. Find the nth-order Taylor polynomials of the given function centered at 0, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = (1 + x)-1/2
a. Find the nth-order Taylor polynomials of the given function centered at 0, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = ln (1 - x)
a. Find the nth-order Taylor polynomials of the given function centered at 0, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = e-x
a. Find the nth-order Taylor polynomials of the given function centered at 0, for n = 0, 1, and 2.b. Graph the Taylor polynomials and the function.f(x) = cos x
a. Find the linear approximating polynomial for the following functions centered at the given point a.b. Find the quadratic approximating polynomial for the following functions centered at the given point a.c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.f(x)
a. Find the linear approximating polynomial for the following functions centered at the given point a.b. Find the quadratic approximating polynomial for the following functions centered at the given point a.c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.f(x)
a. Find the linear approximating polynomial for the following functions centered at the given point a.b. Find the quadratic approximating polynomial for the following functions centered at the given point a.c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.f(x)
a. Find the linear approximating polynomial for the following functions centered at the given point a.b. Find the quadratic approximating polynomial for the following functions centered at the given point a.c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.f(x)
a. Find the linear approximating polynomial for the following functions centered at the given point a.b. Find the quadratic approximating polynomial for the following functions centered at the given point a.c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.f(x)
a. Find the linear approximating polynomial for the following functions centered at the given point a.b. Find the quadratic approximating polynomial for the following functions centered at the given point a.c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.f(x)
a. Find the linear approximating polynomial for the following functions centered at the given point a.b. Find the quadratic approximating polynomial for the following functions centered at the given point a.c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.f(x)
a. Find the linear approximating polynomial for the following functions centered at the given point a.b. Find the quadratic approximating polynomial for the following functions centered at the given point a.c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.f(x)
Explain how to estimate the remainder in an approximation given by a Taylor polynomial.
How is the remainder Rn(x) in a Taylor polynomial defined?
In general, how many terms do the Taylor polynomials p2 and p3 have in common?
The first three Taylor polynomials for f(x) = √1 + x centered at 0 are p0(x) = 1, Find three approximations to √1.1. P1(x) = 1 + 2° and p2(x) = 1 + 2. 8.
Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.
Suppose you use a second-order Taylor polynomial centered at 0 to approximate a function f . What matching conditions are satisfied by the polynomial?
Let an = max {sin 1, sin 2, . . . , sin n}, for n = 1, 2, 3, . . ., where max { . . .} denotes the maximum element of the set. Does {an} converge? If so, make a conjecture about the limit.
The fractal called the Sierpinski triangle is the limit of a sequence of figures. Starting with the equilateral triangle with sides of length 1, an inverted equilateral triangle with sides of length 1/2 is removed. Then, three inverted equilateral triangles with sides of length 1/4 are removed from
Find the limits of the sequences {an} and {bn}.a. 1 || " style="" class="fr-fic fr-dii">b. 1,п > 1 bn " style="" class="fr-fic fr-dii"> x" dx, n > 1 || п "ах р> 1,п > 1 bn
Let {xn} be generated by the recurrence relation x0 = 1 and xn + 1 = xn + cos xn, for n = 0, 1, 2, . . . Use a calculator (in radian mode) to generate as many terms of the sequence {xn} needed to find the integer p such that /p. = TT lim Xn
There is one circle of radius 1. There are two circles of radius 1/2. There are 2n - 1 circles of radius 2-n + 1.a. What is the sum of the areas of the circles on page n of the book?b. Assuming the book continues indefinitely (n→∞), what is the sum of the areas of all the circles in the book?
As in Exercise 65, a crew of workers is constructing a tunnel. The time required to dig 100 m increases by 10% each week, starting with 1 week to dig the first 100 m. Can the crew complete a 1.5-km (1500-m) tunnel in 30 weeks? Explain.Data from Exercise 65A crew of workers is constructing a tunnel
A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose that each week the crew digs 0.95 of the distance it dug the previous week. In the first
Solve the following equations for x.a.b.c. 00 ekr = 2 ,kx k=0 (3x) = 4 k=0
How many terms of the series must be summed to ensure that the error is less than 10-8? +1 k4 k=1
Approximate the series by evaluating the first 20 terms. Compute an upper bound for the error in the approximation. k=1
Approximate the series by evaluating the first 20 terms. Compute an upper bound for the error in the approximation. k=1 5k
Show that the series provided p > 1. 1 k(In k)' k=2
Find the values of p for which converges conditionally. (-1)* kP k=1
Let Rn be the remainder associated withFind an upper bound for Rn (in terms of n). How many terms of the series must be summed to approximate the series with an error less than 10-4? k=1
Let Sn be the nth partial sum of Find 00 Σαι 8. k=1 lim ar and lim S„.
Is it true that the geometric sequence {rk} converges if and only if the geometric seriesconverges? ΣΗ pk k=1
a. Does the sequence converge? Why or why not?b. Does the series converge? Why or why not? k k + 1 k k + 1 k=1
Give an example (if possible) of a series that converges, while the sequence {ak} diverges. Σ α k=1
Give an example (if possible) of a sequence {ak} that converges, while the series diverges. χ Σ ακ k=1
a. Find the limit of the sequenceb. Evaluate 1 k + 1) k ΣΗ k + 1 k=1 8.
a. Find the limit of the sequence {(-4/5)}.b. Evaluate k 4 5 k=0
Determine whether the following series converge or diverge. In the case of convergence, state whether the convergence is conditional or absolute. (-1)* k=0 e^ + e 8.
Determine whether the following series converge or diverge. In the case of convergence, state whether the convergence is conditional or absolute. (-2)k+1 .2 k=1
Determine whether the following series converge or diverge. In the case of convergence, state whether the convergence is conditional or absolute. (-1)* 00 k In k k=2
Determine whether the following series converge or diverge. In the case of convergence, state whether the convergence is conditional or absolute. (-1)*+l10* I+: k! k=1 8.
Determine whether the following series converge or diverge. In the case of convergence, state whether the convergence is conditional or absolute. (-1)* –1) k=1 Vk? + 1
Determine whether the following series converge or diverge. In the case of convergence, state whether the convergence is conditional or absolute. Σ(-1 ket k=1
Determine whether the following series converge or diverge. In the case of convergence, state whether the convergence is conditional or absolute. (-1)*+'(k² + 4) 2k2 + 1 k=1
Determine whether the following series converge or diverge. In the case of convergence, state whether the convergence is conditional or absolute. (-1)* Σ k=2 k- - 1 00 .2
Use a convergence test of your choice to determine whether the following series converge or diverge. 2sech k k=0
Use a convergence test of your choice to determine whether the following series converge or diverge. Etanh k k=1
Use a convergence test of your choice to determine whether the following series converge or diverge. 1 k=1Sinh k
Use a convergence test of your choice to determine whether the following series converge or diverge. 00 coth k k k=1
Use a convergence test of your choice to determine whether the following series converge or diverge. 9k Σ (2k)! k=0
Use a convergence test of your choice to determine whether the following series converge or diverge. 2.4* Σ F (2k + 1)! k=0
Use a convergence test of your choice to determine whether the following series converge or diverge. Σke -k k=1
Use a convergence test of your choice to determine whether the following series converge or diverge. In k² k2 k=1
Use a convergence test of your choice to determine whether the following series converge or diverge. 2 Σ k2 – 10
Use a convergence test of your choice to determine whether the following series converge or diverge. Σκο .5,-k k°e-k k=1
Use a convergence test of your choice to determine whether the following series converge or diverge. 1 + In k k=1
Use a convergence test of your choice to determine whether the following series converge or diverge. Vk k=1 k 3
Use a convergence test of your choice to determine whether the following series converge or diverge. 1 Ek sin k k=1
Use a convergence test of your choice to determine whether the following series converge or diverge. 3 2 + ek k=1
Use a convergence test of your choice to determine whether the following series converge or diverge. 1 =1 VkVk + 1
Use a convergence test of your choice to determine whether the following series converge or diverge. 2*k! .k k=1 8.

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