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mathematics
calculus 10th edition

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

In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) 1 1 - x' c=2
In Exercises match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).](a)(b)(c)(d) 9 an 5 4 3 2- I ترا 2 u ||||||| 4 6 8 10
In Exercises match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).](a)(b)(c)(d) 9 an 5 4 3 2- I ترا 2 u ||||||| 4 6 8 10
Identical blocks of unit length are stacked on top of each other at the edge of a table. The center of gravity of the top block must lie over the block below it, the center of gravity of the top two blocks must lie over the block below them, and so on (see figure).(a) When there are three blocks,
In Exercises write the first five terms of the sequence. an 2n n+ 5
Let T be an equilateral triangle with sides of length 1. Let an be the number of circles that can be packed tightly in n rows inside the triangle. For example, a₁ = 1, a₂ = 3, and a3 = 6, as shown in the figure. Let An be the combined area of the a, circles. Find lim A₁- 11x n-
In Exercises write the first five terms of the sequence. || (一)起 4
In Exercises write the first five terms of the sequence. an || 3n n!
In Exercises write the first five terms of the sequence.an = 5n
The Cantor set (Georg Cantor, 1845-1918) is a subset of the unit interval [0, 1]. To construct the Cantor set, first remove the middle third of the interval, leaving two line segments. For the second step, remove the middle third of each of the two remaining segments, leaving four line
A company buys a machine for $175,000. During the next 5 years, the machine will depreciate at a rate of 30% per year. (That is, at the end of each year, the depreciated value will be 70% of what it was at the beginning of the year.)(a) Find a formula for the nth term of the sequence that gives the
A deposit of $8000 is made in an account that earns 5% interest compounded quarterly. The balance in the account after quarters is(a) Compute the first eight terms of the sequence {An}.(b) Find the balance in the account after 10 years by computing the 40th term of the sequence. An = 8000
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.) 1 2 3 4 2' 5' 10' 17'
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.) 1 1 1 1 1 2' 3' 7' 25' 121'
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.)-5, -2, 3, 10, 19, . . .
In Exercises write an expression for the nth term of the sequence. (There is more than one correct answer.)3, 8, 13, 18, 23, . . .
In Exercises find the first four nonzero terms of the Maclaurin series for the function by multiplying or dividing the appropriate power series. Use the table of power series for elementary functions. Use a graphing utility to graph the function and its corresponding polynomial approximation. x
In Exercises use a power series to approximate the value of the integral with an error of less than 0.0001. Assume that the integrand is defined as 1 when x = 0. (1/2 0 arctan xả dx
Write the power series for (1 + x)k in terms of binomial coefficients.
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = ex²/2
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = In(1 + x²)
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. g(x) = e-3x
In Exercises use the binomial series to find the Maclaurin series for the function. zx + I^ = (x)ƒ
In Exercises use the binomial series to find the Maclaurin series for the function. f(x)=√1 + x³
In Exercises use the binomial series to find the Maclaurin series for the function. f(x) = = 1 2 √4 + x²
In Exercises use the binomial series to find the Maclaurin series for the function. f(x) = = 1 1- x
In Exercises use the binomial series to find the Maclaurin series for the function. f(x) = 4/1 + x
In Exercises use the binomial series to find the Maclaurin series for the function. f(x) = √1 + x
In Exercises use the binomial series to find the Maclaurin series for the function. f(x) 1 = (2 + x)³
In Exercises use the binomial series to find the Maclaurin series for the function. f(x) = 1 (1 + x)²
In Exercises use the binomial series to find the Maclaurin series for the function. f(x) 1 √1-x
In Exercises use the binomial series to find the Maclaurin series for the function. f(x) 1 (1 + x)4
In Exercises prove that the Maclaurin series for the function converges to the function for all x. f(x) = COS X
In Exercises prove that the Maclaurin series for the function converges to the function for all x. f(x) = cosh x
In Exercises prove that the Maclaurin series for the function converges to the function for all x. f(x) = sinh x
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = ln(x² + 1), c = 0
In Exercises prove that the Maclaurin series for the function converges to the function for all x. f(x) = e-2x
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = = tan x, c = 0 (first three nonzero terms)
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = sec x, c = 0 (first three nonzero terms)
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = 1/2/3/ X c = 1
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = sin x, c C = πT 4
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = = sin 3x, c = 0
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = ex, c = 1
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = ln x, c = 1
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = COS X, C = 4
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = e-4x₂ c = 0
Prove that e is irrational. e = 1 + 1 + — + · · · + — + · · · 2! n!
In Exercises use the definition of Taylor series to find the Taylor series, centered at c for the function. f(x) = ²x, c = 0 e2r
In Exercises evaluate the binomial coefficient using the formulawhere k is a real number, n is a positive integer, and (k)= = k(k-1)(k − 2)(k − 3) (kn + 1) - n! .
Show that the Maclaurin series for the function g(x) is 00 n=1 = X 1 - x - x² F₁x n where F is the nth Fibonacci number with F₁ = F₂ = 1 and F₁ = Fn-2+ F-1, for n ≥ 3. Fn Write X ao + ax + a₂x² + 1- x - x² and multiply each side of this equation by 1 - x - x².)
Assume that |ƒ(x)| ≤ 1 and [ƒ"(x)| ≤ 1 for all x on an interval of length at least 2. Show that |ƒ'(x)| ≤ 2 on theinterval.
In Exercises evaluate the binomial coefficient using the formulawhere k is a real number, n is a positive integer, and (k)= = k(k-1)(k − 2)(k − 3) (kn + 1) - n! .
In Exercises evaluate the binomial coefficient using the formulawhere k is a real number, n is a positive integer, and (k)= = k(k-1)(k − 2)(k − 3) (kn + 1) - n! .
Find the Maclaurin series forand determine its radius of convergence. Use the first four terms of the series to approximate In 3. f(x) = In 1 + x 1 - x
In Exercises evaluate the binomial coefficient using the formulawhere k is a real number, n is a positive integer, and (k)= = k(k-1)(k − 2)(k − 3) (kn + 1) - n! .
Use the result of Exercise 83 to determine the series for the path of a projectile launched from ground level at an angle of θ = 60°, with aninitial speed of v0 = 64 feet per second and a drag factor of k = 1/16.Data from in Exercise 83A projectile fired from the ground follows the trajectory
Explain how to use the seriesto find the series for each function. Do not find the series.(a)(b)(c) g(x) = e = n=0 xn n!
Prove thatfor any real x. xn lim = 0 n→∞0 n!
In Exercises use a computer algebra system to find the fifth-degree Taylor polynomial, centered at c, for the function. Graph the function and the polynomial. Use the graph to determine the largest interval on which the polynomial is a reasonable approximation of the function. h(x) = 3 x
A projectile fired from the ground follows the trajectory given bywhere v0 is the initial speed, θ is the angle of projection, g is the acceleration due to gravity, and k is the drag factor caused by air resistance. Using the power series representationverify that the trajectory can be rewritten
In Exercises use a computer algebra system to find the fifth-degree Taylor polynomial, centered at c, for the function. Graph the function and the polynomial. Use the graph to determine the largest interval on which the polynomial is a reasonable approximation of the function. g(x) =
In Exercises use a computer algebra system to find the fifth-degree Taylor polynomial, centered at c, for the function. Graph the function and the polynomial. Use the graph to determine the largest interval on which the polynomial is a reasonable approximation of the function. f(x) = x cos 2x, c = 0
In Exercises use a computer algebra system to find the fifth-degree Taylor polynomial, centered at c, for the function. Graph the function and the polynomial. Use the graph to determine the largest interval on which the polynomial is a reasonable approximation of the function. f(x) = sin In(1
In Exercises use a power series to approximate the value of the integral with an error of less than 0.0001. Assume that the integrand is defined as 1 when x = 0. 0.2 √1 + x² dx
In Exercises approximate the normal probability with an error of less than 0.0001, where the probability is given byP(1 < x < 2) P(a < x < b) = 1 √√2T b Se e-x²/2 dx.
Define the binomial series. What is its radius of convergence?
In Exercises approximate the normal probability with an error of less than 0.0001, where the probability is given byP(0 < x < 1) P(a < x < b) = 1 √√2T b Se e-x²/2 dx.
State the guidelines for finding a Taylor series.
In Exercises use a power series to approximate the area of the region. Use a graphing utility to verify the result. J0.5 y of 1.5 1.0 cos √x dx 0.5 0.5 111 1 1.5 X
In Exercises use a power series to approximate the area of the region. Use a graphing utility to verify the result. *π/2 314112 14 √x cos x dx K00 R|4 - 500 00
In Exercises use a power series to approximate the value of the integral with an error of less than 0.0001. Assume that the integrand is defined as 1 when x = 0. 1/4 x In(x + 1) dx
In Exercises use a power series to approximate the value of the integral with an error of less than 0.0001. Assume that the integrand is defined as 1 when x = 0. 0.3 SE 0.1 √1 + x³ dx
In Exercises use a power series to approximate the value of the integral with an error of less than 0.0001. Assume that the integrand is defined as 1 when x = 0. 1/2 0 arctan x X - dx
In Exercises use a power series to approximate the value of the integral with an error of less than 0.0001. Assume that the integrand is defined as 1 when x = 0. 5 e-x²³ dx
In Exercises use a power series to approximate the value of the integral with an error of less than 0.0001. Assume that the integrand is defined as 1 when x = 0. Jo cos x² dx
In Exercises use a power series to approximate the value of the integral with an error of less than 0.0001. Assume that the integrand is defined as 1 when x = 0. So sin x X dx
In Exercises use the series representation of the function ƒ to find(if it exists). lim f(x) x-0
In Exercises use the series representation of the function ƒ to find(if it exists). lim f(x) x-0
In Exercises verify the sum. Then use a graphing utility to approximate the sum with an error of less than 0.0001. Î (−1)n-1 n=1 n! e-1 e
In Exercises use the series representation of the function ƒ to find(if it exists). lim f(x) x-0
In Exercises verify the sum. Then use a graphing utility to approximate the sum with an error of less than 0.0001. 2" n=on! = e²
In Exercises use the series representation of the function ƒ to find(if it exists). lim f(x) x-0
In Exercises verify the sum. Then use a graphing utility to approximate the sum with an error of less than 0.0001. n=0 (−1)² 1) ² 1 (2n + 1)! = sin 1
In Exercises find a Maclaurin series for ƒ(x). f(x) = S₁ 0 √1 + 1³ dt
In Exercises verify the sum. Then use a graphing utility to approximate the sum with an error of less than 0.0001. Ž (-1)^²+1 -¹ = In 2 n n=1
In Exercises find a Maclaurin series for ƒ(x). =f² 0 f(x) = = (e-t²-1)dt
In Exercises find the first four nonzero terms of the Maclaurin series for the function by multiplying or dividing the appropriate power series. Use the table of power series for elementary functions. Use a graphing utility to graph the function and its corresponding polynomial approximation.
In Exercises find the first four nonzero terms of the Maclaurin series for the function by multiplying or dividing the appropriate power series. Use the table of power series for elementary functions. Use a graphing utility to graph the function and its corresponding polynomial approximation. f(x)
In Exercises find the first four nonzero terms of the Maclaurin series for the function by multiplying or dividing the appropriate power series. Use the table of power series for elementary functions. Use a graphing utility to graph the function and its corresponding polynomial approximation.
In Exercises use a power series and the fact that t² = -1 to verify the formula. 1 g(x) = (eix + e-ix) = COS X cos
In Exercises find the first four nonzero terms of the Maclaurin series for the function by multiplying or dividing the appropriate power series. Use the table of power series for elementary functions. Use a graphing utility to graph the function and its corresponding polynomial approximation. f(x)
In Exercises find the first four nonzero terms of the Maclaurin series for the function by multiplying or dividing the appropriate power series. Use the table of power series for elementary functions. Use a graphing utility to graph the function and its corresponding polynomial approximation. h(x)
In Exercises find the Maclaurin series for the function. f(x) arcsin x, X 1, x #0 x = 0
In Exercises find the Maclaurin series for the function. g(x) = (sin x, X 1, x = 0 x = 0
In Exercises use a power series and the fact that t² = -1 to verify the formula. g(x) (eixe-ix) = sin x 2i
In Exercises find the Maclaurin series for the function. h(x) = x cos x
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = ex + e¯x = 2 cosh x
In Exercises find the Maclaurin series for the function. f(x) = x sin x
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = sinh−1 x = ln(x + v x2 + 1)
In Exercises find the Maclaurin series for the function. Use the table of power series for elementary functions. f(x) = cos²x

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