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study help
mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Explain the Cartesian-to-polar method for graphing polar curves.
Explain three symmetries in polar graphs and how they are detected in equations.
What is the polar equation of the horizontal line y = 5?
What is the polar equation of the vertical line x = 5?
What is the polar equation of a circle of radius |a| centered at the origin?
Write the equations that are used to express a point with Cartesian coordinates (x, y) in polar coordinates.
Write the equations that are used to express a point with polar coordinates (r, θ) in Cartesian coordinates.
Plot the points with polar coordinates (2, π/6) and (-3, -π/2). Give two alternative sets of coordinate pairs for both points.
a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series.c. Give the interval of convergence of the power series for the derivative.f(x) = ex
Use the Ratio or Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate. (1 — 2x)-1/2 et lim х—0 .2 8x2
Use the Ratio or Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate. ,-2x – 4 e-x/2 + 3 lim 2.x2 х>0
Use the Ratio or Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate. lim x(e/* – 1)
Use the Ratio or Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate. х — 2 lim х—2 In (x — 1)
Use the Ratio or Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate. х lim x→1 In x
Use the Ratio or Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate. 12x – 8x3 lim &r3 6 sin 2x
Use the Ratio or Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate. Vī + x – 1 – (x/2) lim 4x2
Use the Ratio or Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate. 3 tan x — 3х + x* lim -1 х—0
Find the remainder term Rn(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)f(x) = ln (1 - x); bound R3(x), for |x| < 1/2.
Find the remainder term Rn(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)f(x) = sin x; bound R3(x), for |x| < π.
Find the remainder term Rn(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)f(x) = ex; bound R3(x), for |x| < 1.
a. Find the Taylor polynomials of order n = 0, 1, and 2 for the given functions centered at the given point a.b. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.f(x) = sin x, a = π/4; approximate sin (π/5).
a. Find the Taylor polynomials of order n = 0, 1, and 2 for the given functions centered at the given point a.b. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.f(x) = √1 + x, a = 0; approximate √1.08.
a. Find the Taylor polynomials of order n = 0, 1, and 2 for the given functions centered at the given point a.b. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.f(x) = ex, a = 0; approximate e-0.08.
a. Find the Taylor polynomials of order n = 0, 1, and 2 for the given functions centered at the given point a.b. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.f(x) = cos x, a = 0; approximate cos (-0.08).
Find the nth-order Taylor polynomial for the following functions centered at the given point a.f(x) = cosh x, n = 3, a = ln 2
Find the nth-order Taylor polynomial for the following functions centered at the given point a.f(x) = sinh 2x, n = 4, a = 0
Find the nth-order Taylor polynomial for the following functions centered at the given point a.f(x) = ln x, n = 2, a = 1
Find the nth-order Taylor polynomial for the following functions centered at the given point a.f(x) = cos x, n = 2, a = π/4
Find the nth-order Taylor polynomial for the following functions centered at the given point a.f(x) = ln (1 + x), n = 3, a = 0
Find the nth-order Taylor polynomial for the following functions centered at the given point a.f(x) = e-x, n = 2, a = 0
Find the nth-order Taylor polynomial for the following functions centered at the given point a.f(x) = cos x2, n = 2, a = 0
Identify the functions represented by the following power series. ο0 μ4 Σ-1) 3k k k=0
Let f(x) = (ex - 1)/x, for x ≠ 0, and f(0) = 1. Use the Taylor series for f and f' about 0 to evaluate f'(2) to find the value of
Let f(x) = (ex - 1)/x, for x ≠ 0, and f(0) = 1. Use the Taylor series for f about 0 and evaluate f(1) to find the value of
Find the remainder in the Taylor series centered at the point a for the following functions. Then show thatfor all x in the interval of convergence.f(x) = cos x, a = π/2 lim R„(x) = 0
a. Find the first four nonzero terms of the Taylor series for the given function centered at a.b. Write the power series using summation notation.f(x) = 1/x, a = 2
Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10-4 in magnitude. Although you do not need it, the exact value of the series is given in each case. (-1)* Σ π 4 =o 2k + 1 8.
Complete the following steps to find the values of p > 0 for which the series converges.a. Use the Ratio Test to show that convergesfor p > 2.b. Use Stirling’s formula, k! ≈ √2k kke-k for large k, to determine whether the series converges when p = 2. 1•3•5•· · (2k – 1) p*k!
Working in the early 1600s, the mathematicians Wallis, Pascal, and Fermat were calculating the area of the region under the curve y = xp between x = 0 and x = 1, where p is a positive integer. Using arguments that predated the Fundamental Theorem of Calculus, they were able to prove thatUse what
Use the ideas of Exercise 88 to evaluate the following infinite products.a.b. = e•e!/2• el/4• e!/8.. k=0 1 2 3 4 П k 2 3 4 5 k=2
An infinite product P = a1 a2 a3 . . , which is denoted is the limit of the sequence of partial products {a1, a1 a2, a1 a2 a3, . . .}. Assume that ak > 0 for all k.a. Show that the infinite product converges (which means its sequence of partial products converges) provided the series
Use the Ratio Test to determine the values of x ≥ 0 for which each series converges. 2k k=1
Use the Ratio Test to determine the values of x ≥ 0 for which each series converges. 2k k2 そマ k=1 8.
Use the Ratio Test to determine the values of x ≥ 0 for which each series converges. .2 k=1 そマ
Use the Ratio Test to determine the values of x ≥ 0 for which each series converges. 00 k k=1
Use the Ratio Test to determine the values of x ≥ 0 for which each series converges. Σ. k k=1
Use the Ratio Test to determine the values of x ≥ 0 for which each series converges. k! k=1 8.
Use the proof of case (1) of the Limit Comparison Test (Theorem 8.17) to prove cases (2) and (3).
Determine whether the following series converge.a.b. Σsin k k=1 1 sin k k=1k
We know from Section 8.3 that the geometric series ∑ark (a ≠ 0) converges if 0 < r < 1 and diverges if r > 1. Prove these facts using the Integral Test, the Ratio Test, and the Root Test. Now consider all values of r. What can be determined about the geometric series using the
Prove that if ∑ak is a convergent series of positive terms, then the series∑a2k also converges.
Find the values of the parameter p > 0 for which the following series converge. k k
Find the values of the parameter p > 0 for which the following series converge. ο0 Ση k + 1 k=1
Find the values of the parameter p > 0 for which the following series converge. kpk Σ k + 1 k=1
Find the values of the parameter p > 0 for which the following series converge.Stirling’s formula is useful: k! ≈ √2πk kke-k for large k. k! p* k=0 (k + 1)k
Find the values of the parameter p > 0 for which the following series converge. In k\P k=2
Find the values of the parameter p > 0 for which the following series converge.
Find the values of the parameter p > 0 for which the following series converge. In k kP k=2
Find the values of the parameter p > 0 for which the following series converge. 1 (In k)º k=2
Use the test of your choice to determine whether the following series converge. 4 9. 16 1! 4! 2! 3!
Use the test of your choice to determine whether the following series converge. 2 3 22 32 42
Use the test of your choice to determine whether the following series converge. 1 5• 7 1:3 3.5
Use the test of your choice to determine whether the following series converge. ο0 Σ100k k=2
Use the test of your choice to determine whether the following series converge. 1 tan k Σ k=1
Use the test of your choice to determine whether the following series converge. 1 S1 k Σ sin? k=1
Use the test of your choice to determine whether the following series converge. In k k k=2
Use the test of your choice to determine whether the following series converge. Ek-1/k k=1
Use the test of your choice to determine whether the following series converge. k + 2 Ση k + 1 k=1
Use the test of your choice to determine whether the following series converge. k² In k k=2
Use the test of your choice to determine whether the following series converge. 00 ΣΑ p > 0 k!- kl+p* k=1
Use the test of your choice to determine whether the following series converge. 00 (1 + p)k k=1
Use the test of your choice to determine whether the following series converge.
Use the test of your choice to determine whether the following series converge.
Use the test of your choice to determine whether the following series converge. 2k k! k=1
Use the test of your choice to determine whether the following series converge. 00 5 In k k k=2
Use the test of your choice to determine whether the following series converge. + 2-k k=1
Use the test of your choice to determine whether the following series converge. (k!)³ (3k)! k=1
Use the test of your choice to determine whether the following series converge. 1 =1 Vk³ – k + 1
Use the test of your choice to determine whether the following series converge. 5k – 3k k=3
Use the test of your choice to determine whether the following series converge. 1 In k k=3
Use the test of your choice to determine whether the following series converge. A 5k – 1 k=1
Use the test of your choice to determine whether the following series converge. k2 + 2k + 1 3k2 + 1 k=1 И
Use the test of your choice to determine whether the following series converge. 2k Σ k – 1 k=1 e'
Use the test of your choice to determine whether the following series converge. E(Vk – 1)2% k=1
Use the test of your choice to determine whether the following series converge. sin? k S1 k² k=1 8.
Use the test of your choice to determine whether the following series converge. k100 (k + 1)! k=1
Use the test of your choice to determine whether the following series converge. k2 k 2k2 + 1 k=1
Use the test of your choice to determine whether the following series converge. 2 k k=1 8.
Use the test of your choice to determine whether the following series converge. ) - () - () 2 3 4 3 3
Determine whether the following statements are true and give an explanation or counterexample.a. Suppose that 0 < ak < bk. If ∑ak converges, then ∑bk converges.b. Suppose that 0 < ak < bk. If ∑ak diverges, then ∑bk diverges.c. Suppose 0 < bk < ck < ak. If ∑ak
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. Σ (k In k)² k=2
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. Vk? + 1 .2 00 V k³ + 2 k=1
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 1 kVk + 2 k=1
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. Vk k=1 2k – 8.
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 1 3k – 2k k=1
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. sin (1/k) k² k=1 8.
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. k .3 k + 1 k=1
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 1 3, k=1 k/2 + 1
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 0.0001 Σ k + 4 k=1 8.
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. k² – 1 .3 k=1 k + 4 8.
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