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Calculus Volume 2

Review Exercises

Calculus Volume 2Review Exercises

Review Exercises

True or False? In the following exercises, justify your answer with a proof or a counterexample.

253.

If the radius of convergence for a power series n=0anxnn=0anxn is 5,5, then the radius of convergence for the series n=1nanxn1n=1nanxn1 is also 5.5.

254.

Power series can be used to show that the derivative of exisex.exisex. (Hint: Recall that ex=n=01n!xn.)ex=n=01n!xn.)

255.

For small values of x,sinxx.x,sinxx.

256.

The radius of convergence for the Maclaurin series of f(x)=3xf(x)=3x is 3.3.

In the following exercises, find the radius of convergence and the interval of convergence for the given series.

257.

n = 0 n 2 ( x 1 ) n n = 0 n 2 ( x 1 ) n

258.

n = 0 x n n n n = 0 x n n n

259.

n = 0 3 n x n 12 n n = 0 3 n x n 12 n

260.

n = 0 2 n e n ( x e ) n n = 0 2 n e n ( x e ) n

In the following exercises, find the power series representation for the given function. Determine the radius of convergence and the interval of convergence for that series.

261.

f ( x ) = x 2 x + 3 f ( x ) = x 2 x + 3

262.

f ( x ) = 8 x + 2 2 x 2 3 x + 1 f ( x ) = 8 x + 2 2 x 2 3 x + 1

In the following exercises, find the power series for the given function using term-by-term differentiation or integration.

263.

f ( x ) = tan −1 ( 2 x ) f ( x ) = tan −1 ( 2 x )

264.

f ( x ) = x ( 2 + x 2 ) 2 f ( x ) = x ( 2 + x 2 ) 2

In the following exercises, evaluate the Taylor series expansion of degree four for the given function at the specified point. What is the error in the approximation?

265.

f ( x ) = x 3 2 x 2 + 4 , a = −3 f ( x ) = x 3 2 x 2 + 4 , a = −3

266.

f ( x ) = e 1 / ( 4 x ) , a = 4 f ( x ) = e 1 / ( 4 x ) , a = 4

In the following exercises, find the Maclaurin series for the given function.

267.

f ( x ) = cos ( 3 x ) f ( x ) = cos ( 3 x )

268.

f ( x ) = ln ( x + 1 ) f ( x ) = ln ( x + 1 )

In the following exercises, find the Taylor series at the given value.

269.

f ( x ) = sin x , a = π 2 f ( x ) = sin x , a = π 2

270.

f ( x ) = 3 x , a = 1 f ( x ) = 3 x , a = 1

In the following exercises, find the Maclaurin series for the given function.

271.

f ( x ) = e x 2 1 f ( x ) = e x 2 1

272.

f ( x ) = cos x x sin x f ( x ) = cos x x sin x

In the following exercises, find the Maclaurin series for F(x)=0xf(t)dtF(x)=0xf(t)dt by integrating the Maclaurin series of f(x)f(x) term by term.

273.

f ( x ) = sin x x f ( x ) = sin x x

274.

f ( x ) = 1 e x f ( x ) = 1 e x

275.

Use power series to prove Euler’s formula: eix=cosx+isinxeix=cosx+isinx

The following exercises consider problems of annuity payments.

276.

For annuities with a present value of $1$1 million, calculate the annual payouts given over 2525 years assuming interest rates of 1%,5%,and10%.1%,5%,and10%.

277.

A lottery winner has an annuity that has a present value of $10$10 million. What interest rate would they need to live on perpetual annual payments of $250,000?$250,000?

278.

Calculate the necessary present value of an annuity in order to support annual payouts of $15,000$15,000 given over 2525 years assuming interest rates of 1%,5%,and10%.1%,5%,and10%.

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