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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Suppose f and g have Taylor series about the point a. a. If f(a) = g(a) = 0 and g'(a) ≠ 0, evaluate by expanding f and g in their Taylor series. Show that the result is consistent with l’Hôpital’s Rule.b. If f(a) = g(a) = f'(a) = g'(a) = 0 and g"(a) ≠ 0, evaluate by expanding f and g
We know thatUse long division to determine exactly how csc x grows as x→0+. Specifically, find a, b, and c (all positive) in the following sentence: As x→0+, csc x ≈ a/xb + cx. lim csc x = 00 r>0+
a. Use infinite series to show that cos x is an even function. That is, show cos (-x) = cos x.b. Use infinite series to show that sin x is an odd function. That is, show sin (-x) = -sin x.
Use the identity sec x = 1/cos x and long division to find the first three terms of the Maclaurin series for sec x.
Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions isa. Write out the first four terms of J0.b. Find the radius and interval of
An essential function in statistics and the study of the normal distribution is the error functiona. Compute the derivative of erf (x).b. Expand e-t2 in a Maclaurin series; then integrate to find the first four nonzero terms of the Maclaurin series for erf.c. Use the polynomial in part (b) to
The theory of optics gives rise to the two Fresnel integralsa. Compute S'(x) and C'(x).b. Expand sin t2 and cos t2 in a Maclaurin series and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.c. Use the polynomials in part (b) to approximate S(0.05) and
The function is called the sine integral function.a. Expand the integrand in a Taylor series about 0.b. Integrate the series to find a Taylor series for Si.c. Approximate Si(0.5) and Si(1). Use enough terms of the series so the error in the approximation does not exceed 10-3. sin t dt Si(x) =
The period of a pendulum is given bywhere ℓ is the length of the pendulum, g ≈ 9.8 m/s2 is the acceleration due to gravity, k = sin (θ0/2), and θ0 is the initial angular displacement of the pendulum (in radians). The integral in this formula F(k) is called an elliptic integral, and it cannot
Teams A and B go into sudden death overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Each team has a 1/6 chance of scoring when it has the ball, with Team A having the ball first. a. The probability that Team A ultimately wins is
The expected (average) number of tosses of a fair coin required to obtain the first head is Evaluate this series and determine the expected number of tosses. Differentiate a geometric series. ΣΚ 3. k=1
Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Explain why f(k) (a) = k! multiplied by the coefficient of (x - a)k.
Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Explain why f(k) (a) = k! multiplied by the coefficient of (x - a)k.
Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Explain why f(k) (a) = k! multiplied by the coefficient of (x - a)k.
Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Explain why f(k) (a) = k! multiplied by the coefficient of (x - a)k.
The inverse hyperbolic sine is defined in several ways; among them areFind the first four terms of the Taylor series for sinh-1x using these two definitions. dt sinhx = In (x + Vx² + 1) V1 + t²
Use Taylor series to evaluate sin x \/x² lim х>0 х
Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). tan -1 ах sin ax – lim aх bx³
Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). sin ax lim x→0 sin bx S1
Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). ear lim ах - 1 х>0 х
Determine whether the following statements are true and give an explanation or counterexample.a. To evaluate one could expand the integrand in a Taylor series and integrate term by term.b. To approximate π/3, one could substitute x = √3 into the Taylor series for tan-1 x.c. dx 1 – x' (In 2)*
Identify the functions represented by the following power series. 00 – 1) k=2 k(k – 1)
Identify the functions represented by the following power series. k(k – 1)x* 3k k=2
Identify the functions represented by the following power series. 2k k k=1
Identify the functions represented by the following power series. ο kxk+1 Σ-1. 3k k=1
Identify the functions represented by the following power series. (-1)* xk+1 4k k=0
Identify the functions represented by the following power series. k k=1
Identify the functions represented by the following power series. Σ2. 2k x2k+1 k=0
Identify the functions represented by the following power series. 2k Σ-1) k=0
Identify the functions represented by the following power series. 2k k=0
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 00 Σ k=1 k• 2k
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 (the alternating harmonic series). (-1)*+1 k k=1
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
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.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.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.√e
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 a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10-4. In (1 + t) dt •0.2 Jo
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 dx V1 + x°
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. 0.4 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. »0.35 tan x dx х ах
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. 0.2 xª dx V1 + Jo
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 cos 2x² dx 2r2 cos -0.35
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. •0.2 sin x? 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. r0.25 dx
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) = 2
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) = 2
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) = 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) - 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 for the derivative.f(x) = -ln (1 - x)
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) = tan-1 x
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) = (1 - x)-1
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) = e-2x
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) = sin x2
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) = ln (1 + x)
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) = cos x
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
Evaluate the following limits using Taylor series. (1 — 2х) -1/2 e* lim х—>0 8x?
Evaluate the following limits using Taylor series. -2x – 4 e-x/2 + 3 lim 2r2 х>0
Evaluate the following limits using Taylor series. lim x(e'/* – 1) 1/x |x→∞
Evaluate the following limits using Taylor series. 2 lim x→2 In (x – 1)
Evaluate the following limits using Taylor series. lim x→1 In x
Evaluate the following limits using Taylor series. 12x – 8x3 lim 6 sin 2x
Evaluate the following limits using Taylor series. VI + x – 1 – (x/2) lim 4x2
Evaluate the following limits using Taylor series. 3 tan x — 3х + х lim х>0 .5 +১
Evaluate the following limits using Taylor series. x² lim х—4 In (x — 3) 16
Evaluate the following limits using Taylor series. In (1 + x) – x + x²/2 lim 43
Evaluate the following limits using Taylor series. lim x sin х х
Evaluate the following limits using Taylor series. 2 + 4x² 2 cos 2x lim 2x4
Evaluate the following limits using Taylor series. 1 + x – e* lim 4x2
Evaluate the following limits using Taylor series. et – e-* lim х>0
Evaluate the following limits using Taylor series. sin 2x S1i lim
Evaluate the following limits using Taylor series. —х — In (1 — х) lim .2 х>0
Evaluate the following limits using Taylor series. -1 tanx lim 3
Evaluate the following limits using Taylor series. et – 1 lim х х>0
What condition must be met by a function f for it to have a Taylor series centered at a?
If and the series converges for |x| < b, what is the power series for f'(x)? .k Ck -Σοgxt f (x) = k=0
Suggest a Taylor series and a method for approximating π.
How would you approximate e-0.6 using the Taylor series for ex?
Explain the method presented in this section for approximating where f has a Taylor series with an interval of convergence centered at a that includes b. f(x) dx.
Explain the strategy presented in this section for evaluating a limit of the form where f and g have Taylor series centered at a. lim f (x)/g(x). х—а
Consider the functiona. Use the definition of the derivative to show that f'(0) = 0.b. Assume the fact that f(k) (0) = 0, for k = 1, 2, 3, . . . (You can write a proof using the definition of the derivative.) Write the Taylor series for f centered at 0.c. Explain why the Taylor series for f
Assume that f has at least two continuous derivatives on an interval containing a with f'(a) = 0. Use Taylor’s Theorem to prove the following version of the Second Derivative Test.a. If f"(x) > 0 on some interval containing a, then f has a local minimum at a.b. If f"(x) < 0 on some interval
Explain why the Mean Value Theorem (Theorem 4.9) is a special case of Taylor’s Theorem.
Suppose you want to approximate 3√128 to within 10-4 of the exact value.a. Use a Taylor polynomial for f(x) = (125 + x)1/3 centered at 0.b. Use a Taylor polynomial for f(x) = x1/3 centered at 125.c. Compare the two approaches. Are they equivalent?
Choose a Taylor series and center point to approximate the following quantities with an error of 10-4 or less.1/4√17
Choose a Taylor series and center point to approximate the following quantities with an error of 10-4 or less.3√83
Choose a Taylor series and center point to approximate the following quantities with an error of 10-4 or less.sin (0.98π)
Choose a Taylor series and center point to approximate the following quantities with an error of 10-4 or less.cos 40°
Use composition of series to find the first three terms of the Maclaurin series for the following functions.a. esin xb. etan xc. √1 + sin2 x
Find the next two terms of the following Taylor series. 1:3•5 1:3 2* + 2-4* .2 X² 2.4.6 .3 x’ +
Find the next two terms of the following Taylor series. V1 + x: 1 + X 1:3 2.4•6 +3 2.4
Find a power series that has (2, 6) as an interval of convergence.
By comparing the first four terms, show that the Maclaurin series for cos2 x can be found (a) By squaring the Maclaurin series for cos x.(b) By using the identity cos2 x = (1 + cos 2x)/2.(c) By computing the coefficients using the definition.
By comparing the first four terms, show that the Maclaurin series for sin2 x can be found (a) By squaring the Maclaurin series for sin x.(b) By using the identity sin2 x = (1 - cos 2x)/2.(c) By computing the coefficients using the definition.
Suppose you want to approximate √72 using four terms of a Taylor series. Compare the accuracy of the approximations obtained using Taylor series for √x centered at 64 and 81.
Show that the first five nonzero coefficients of the Taylor series (binomial series) for f(x) = √1 + 4x about 0 are integers. (In fact, all the coefficients are integers.)
Recall that the Taylor series for f(x) = 1/(1 - x) about 0 is the geometric series Show that this series can also be found as a binomial series. k Σε. k=0
Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number.f(x) = 4√x with a = 16; approximate 4√13.
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