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mathematics
calculus early transcendentals

Questions and Answers of Calculus Early Transcendentals

Explain how a pair of parametric equations generates a curve in the xy-plane.
Give two pairs of parametric equations that generate a circle centered at the origin with radius 6.
In the following exercises, two sequences are given, one of which initially has smaller values, but eventually “overtakes” the other sequence. Find the sequence with the larger growth rate and
In the following exercises, two sequences are given, one of which initially has smaller values, but eventually “overtakes” the other sequence. Find the sequence with the larger growth rate and
In the following exercises, two sequences are given, one of which initially has smaller values, but eventually “overtakes” the other sequence. Find the sequence with the larger growth rate and
For what values of a does the sequence {n!} grow faster than the sequence {nan}? Stirling’s formula is useful: n! ≈ √2pn nne-n, for large values of n.
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
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
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
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
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
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
a. Find a power series for the solution of the following differential equations, subject to the given initial condition.b. Identify the function represented by the power series.y'(t) - y = 0, y(0) = 2
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
a. Find a power series for the solution of the following differential equations, subject to the given initial condition.b. Identify the function represented by the power series.y'(t) + 4y = 8, y(0) =
a. Find a power series for the solution of the following differential equations, subject to the given initial condition.b. Identify the function represented by the power series.y'(t) - 3y = 10, y(0)
a. Find a power series for the solution of the following differential equations, subject to the given initial condition.b. Identify the function represented by the power series.y'(t) = 6y + 9, y(0) =
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. 0 0.25 e-x² е dx
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. 0 0.2 2 sin x² dx
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. 0.35 -0.35 cos 2x² dx
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. 0 0.2 V1 + x² dx
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. 0 0.4 In (1 + x²) dx
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. .0.5 0 dx V1 + x6
Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. Jo 0.2 In (1+1) t dt
Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.e2
Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.√e
Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.cos 2
Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.sin 1
Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.ln 3/2
Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.tan-1 1/2
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 M k=0 1 (k + 1)!
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 k=1 k 2k- (k + 1)!*
Write the Taylor series for f(x) = ln (1 + x) about 0 and find its interval of convergence. Assume the Taylor series converges to f on the interval of convergence. Evaluate f(1) to find the value of
Write the Maclaurin series for f(x) = ln (1 + x) and find the interval of convergence. Evaluate f(-1/2) to find the value of 1 k=1 k•2k*
Identify the functions represented by the following power series. k=0 xk 2k
Identify the functions represented by the following power series. Σ(-1)* k=0 ht 3k
Identify the functions represented by the following power series. Σ(-1)* k=0 .2k X 4k
Identify the functions represented by the following power series. k=0 24 x2k+1
Identify the functions represented by the following power series. k=1 xk k
Identify the functions represented by the following power series. 8 k=0 (−1)xk+ 4k
Identify the functions represented by the following power series. ∑(-1)* k=1 tk+1 35
Identify the functions represented by the following power series. 8 x2k k=1 k
Use the formal definition of the limit of a sequence to prove the following limits. lim n→00 3n² 4n² + 1 || 3 4
Use the formal definition of the limit of a sequence to prove the following limits. lim bn n→∞ = 0, for b > 1
Use the formal definition of the limit of a sequence to prove the following limits. for real numbers b > 0 and c > 0 сп lim n→∞ bn + 1 с Б 6
Use the formal definition of the limit of a sequence to prove the following limits. n lim 2 n→∞ n + 1 0
Determine whether the following statements are true and give an explanation or counterexample. a. If b. If c. The convergent sequences {an} and {bn} differ in their first 100 terms, but an = bn,
Express each sequenceas an equivalent sequence of the form {an} n = 1 {bn}n=3. {2n + 1}=
Express each sequenceas an equivalent sequence of the form {an} n = 1 {bn}n=3. {n² + 6n − 9}%= }n=1
Evaluate the limit of the following sequences or state that the limit does not exist. an = •n S 1 x-² dx X
Evaluate the limit of the following sequences or state that the limit does not exist. an = 75"-1 99n + 5" sin n 8n
Evaluate the limit of the following sequences or state that the limit does not exist. an = tan 10n 10n + 4
Evaluate the limit of the following sequences or state that the limit does not exist. an = COS (0.99") + 7h + 9n 63
Evaluate the limit of the following sequences or state that the limit does not exist. an 4" + 5n! n! + 2n
Evaluate the limit of the following sequences or state that the limit does not exist. an || 6n + 3n 100 6n + n
Evaluate the limit of the following sequences or state that the limit does not exist. an = n ntn 7 n² + n³ ln n 7 8
Evaluate the limit of the following sequences or state that the limit does not exist. An n 7" 7 n 5n in
Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the
Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the
Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the
Consider the geometric series which has the value 1/(1 - r) provided |r| be the sum of the first n terms. The magnitude of the remainder Rn is the error in approximating S by Sn. Show that? S = pk
Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the
Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within 10-6 of the value of the series that is, to ensure |Rn| -6). a. b. Data from Exercise
Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the
Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within 10-6 of the value of the series that is, to ensure |Rn| -6). a. b. Data from Exercise
The sequence {n!} ultimately grows faster than the sequence {bn}, for any b > 1, as n →∞. However, bn is generally greater than n! for small values of n. Use a calculator to determine the
Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within 10-6 of the value of the series that is, to ensure |Rn| -6). a. b. Data from Exercise
A fishery manager knows that her fish population naturally increases at a rate of 1.5% per month, while 80 fish are harvested each month. Let Fn be the fish population after the nth month, where F0 =
Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within 10-6 of the value of the series that is, to ensure |Rn| -6). a. b. Data from Exercise
A heifer weighing 200 lb today gains 5 lb per day with a food cost of 45¢/day. The price for heifers is 65¢/lb today but is falling 1¢/day.a. Let hn be the profit in selling the heifer on the nth
Suppose a function f is defined by the geometric series a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5), if possible. b. What is the domain of f ? f(x) = Co x k k=0
After many nights of observation, you notice that if you oversleep one night, you tend to undersleep the following night, and vice versa. This pattern of compensation is described by the
Suppose a function f is defined by the geometric series a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5), if possible. b. What is the domain of f ? f(x) = Σ(-1)*x k=0
The CORDIC (COordinate Rotation DIgital Calculation) algorithm is used by most calculators to evaluate trigonometric and logarithmic functions. An important number in the CORDIC algorithm, called the
Suppose a function f is defined by the geometric series a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5), if possible. b. What is the domain of f? f(x) Σχ k=0 x 2k
For what values of x does the geometric series converge? Solve f(x) = 3. f(x) = k=0 k 1 ( ₁ + x) ² 1
Consider the expression where the process continues indefinitely. a. Show that this expression can be built in steps using the recurrence relation a0 = 1, an + 1 = ?1 + an, for n = 0, 1, 2, 3, ?. . .
Find the limit of the sequence {an}"=2 = {(₁ - 1) (¹ - 3 ) · · · ( ¹ - -)}L, 1 2 n n=2
Imagine a stack of hemispherical soap bubbles with decreasing radii r1 = 1, r2, r3, . . . (see figure). Let hn be the distance between the diameters of bubble n and bubble n + 1, and let Hn be the
The expression where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation a0 = 1, an + 1 = 1 + 1/an ,
Consider the geometric series? a. Fill in the following table that shows the value of the series f(r) for various values of r. b. Graph f, for |r| c. Evaluate OC f(r) = r, where |r| < 1. k=0 r
For a positive real number p, the tower of exponents ppp a continues indefinitely and the expression is ambiguous. The tower could be built from the top as the limit of the sequence {pp, (pp)p,
Consider serieswhere |r| a. Complete the following table showing the smallest value of n, calling it N(r), such that |S - Sn| -4, for various values of r. For example, with r = 0.5 and S = 2, we
The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about a.d. 1200 as a model for the growth of rabbit populations. It is given by the recurrence relation fn +
Pick two positive numbers a0 and b0 with a0 > b0, and write out the first few terms of the two sequences {an} and {bn}: Recall that the arithmetic mean A = (p + q)/2 and the geometric mean G =
Here is a fascinating (unsolved) problem known as the hailstone problem (or the Ulam Conjecture or the Collatz Conjecture). It involves sequences in two different ways. First, choose a positive
Prove that if {an} << {bn} (as used in Theorem 8.6), then {can} << {dbn}, where c and d are positive real numbers.
Consider the sequence defined by? an + 1 = ?3an, a1 = ?3, for n ? 1. a. Show that {an} is increasing. b. Show that {an} is bounded between 0 and 3.? c. Explain whyexists. d. Find? lim an n→∞ lim
In the following exercises, two sequences are given, one of which initially has smaller values, but eventually “overtakes” the other sequence. Find the sequence with the larger growth rate and
In the following exercises, two sequences are given, one of which initially has smaller values, but eventually “overtakes” the other sequence. Find the sequence with the larger growth rate and
Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility.an = (-1)n n√/n
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 0.1 = 0.111...
Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. An = cot NTT 2n + 2
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 0.5 = 0.555..
Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.{0.2n}
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 0.09 = 0.090909...
Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.{1.2n}
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 0.27 0.272727...
Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.{(-0.7)n}
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 0.037 = 0.037037...

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