BC Calc Extended Response – Taylor and Power Series

1. (1971 BC 4) N0 CALCULATOR

(a) Write the first three nonzero terms in the Taylor series expansion of cos x about .

(b) What is the interval of convergence of the Taylor series mentioned in part (a)? Show your method.

(c) Use the first two nonzero terms of the series in part (a) to approximate .

(d) Estimate the accuracy of the approximation found in part (c).Show your method.

2. (1975 BC 4) NO CALCULATOR

(a) Determine whether the series is convergent. Justify your answer.

(b) Find the interval of convergence for the series . Justify your answer.

3. (1976 BC 7)NO CALCULATOR

(a) Write the first three nonzero terms and the general term of the Taylor series expansion of about x = 0.

(b) What is the interval of convergence for the series found in part (a)? Justify your answer.

4. (1978 BC 5)NO CALCULATOR

The power series has the interval of convergence . Let f(x) be its sum.

(a) Find f(0) and f’(0).

(b) Justify the interval of convergence is .

5. (1979 BC 4)NO CALCULATOR

Let f be the function defined by .

(a)Write the first four terms and the general term of the Taylor series expansion of f(x) about x = 0.

(b) What is the interval of convergence for the series found in part (a)? Show your method.

(c) Find the value or f at . How many terms of the series are adequate for approximating with an error not exceeding one per cent? Justify your answer.

6. ( 1982 BC 5) NO CALCULATOR

(a) Write the Taylor series expansion about x = 0 for . Include an expression for the general term.

(b) For what value of x does the series in part (a) converge?

(c) Estimate the error in evaluating by using only the first five nonzero terms of the series in part (a). Justify your answer.

(d) Use the result of part (a) to determine the logarithmic function whose Taylor series is .

7. (1983 BC 5) SCIENTIFIC CALCULATOR

Consider the power series , where and for .

(a) Find the first four terms and the general term of the series.

(b) For what values of x does the series converge?

(c) If , find the value of f’(1).

8. (1984 BC 4)SCIENTIFIC CALCULATOR

Let f be the function defined by for all values or x for which the series converges.

(a) Find the radius of convergence of the series.

(b) Use the first three terms of this series to find an approximation of f(-1).

(c) Estimate the amount of error involved in the approximation in part (b). Justify your answer.

9. (1986 BC 5) NO CALCULATOR

(a) Find the first four nonzero terms in the Taylor series expansion about x = 0 for .

(b) Use the results of part (a) to find the first four nonzero terms in the Taylor series expansion about x = 0 for about x = 0.