Math: Taylor and Maclaurin Series Worksheet

Question 1

Write the first four nonzero terms of the Maclaurin series for e^x.

Accepted Answer

The Maclaurin series for e^x is 1 + x + x^2/2! + x^3/3! + ... . The first four nonzero terms are 1, x, x^2/2, and x^3/6.

Question 2

Write the first four nonzero terms of the Maclaurin series for sin x.

Accepted Answer

The Maclaurin series for sin x is x - x^3/3! + x^5/5! - x^7/7! + ... . The first four nonzero terms are x, -x^3/6, x^5/120, and -x^7/5040.

Question 3

Write the first four nonzero terms of the Maclaurin series for cos x.

Accepted Answer

The Maclaurin series for cos x is 1 - x^2/2! + x^4/4! - x^6/6! + ... . The first four nonzero terms are 1, -x^2/2, x^4/24, and -x^6/720.

Question 4

Use the Maclaurin polynomial 1 + x + x^2/2 to approximate e^0.2. Round your answer to four decimal places.

Accepted Answer

Substituting x = 0.2 gives 1 + 0.2 + (0.2)^2/2 = 1.2 + 0.02 = 1.2200. The approximation is e^0.2 is about 1.2200.

Question 5

Use the Maclaurin polynomial x - x^3/6 to approximate sin(0.3). Round your answer to four decimal places.

Accepted Answer

Substituting x = 0.3 gives 0.3 - (0.3)^3/6 = 0.3 - 0.027/6 = 0.3 - 0.0045 = 0.2955. The approximation is sin(0.3) is about 0.2955.

Question 6

Find the third-degree Taylor polynomial for f(x) = ln x centered at x = 1.

Accepted Answer

For f(x) = ln x, the derivatives at x = 1 are f(1) = 0, f'(1) = 1, f''(1) = -1, and f'''(1) = 2. The third-degree Taylor polynomial is (x - 1) - (x - 1)^2/2 + (x - 1)^3/3.

Question 7

Find the second-degree Taylor polynomial for f(x) = sqrt(x) centered at x = 4.

Accepted Answer

For f(x) = sqrt(x), f(4) = 2, f'(x) = 1/(2sqrt(x)), so f'(4) = 1/4, and f''(x) = -1/(4x^(3/2)), so f''(4) = -1/32. The second-degree Taylor polynomial is 2 + (x - 4)/4 - (x - 4)^2/64.

Question 8

The Maclaurin series for 1/(1 - x) is 1 + x + x^2 + x^3 + ... . Use the first four terms to approximate 1/(1 - 0.1).

Accepted Answer

Using the first four terms gives 1 + 0.1 + 0.1^2 + 0.1^3 = 1 + 0.1 + 0.01 + 0.001 = 1.111. The approximation is 1/(1 - 0.1) is about 1.111.

Question 9

For the geometric series 1 + x + x^2 + x^3 + ... , state the interval of convergence.

Accepted Answer

The geometric series 1 + x + x^2 + x^3 + ... converges when |x| < 1. Its interval of convergence is -1 < x < 1.

Question 10

Use the first three nonzero terms of the Maclaurin series for cos x to approximate cos(0.5). Round your answer to four decimal places.

Accepted Answer

The first three nonzero terms are 1 - x^2/2! + x^4/4!. Substituting x = 0.5 gives 1 - 0.25/2 + 0.0625/24 = 1 - 0.125 + 0.002604... = 0.877604... . The approximation is cos(0.5) is about 0.8776.

Question 11

Find the coefficient of x^5 in the Maclaurin series for sin x.

Accepted Answer

The Maclaurin series for sin x is x - x^3/3! + x^5/5! - x^7/7! + ... . The coefficient of x^5 is 1/5!, which equals 1/120.

Question 12

Explain why a Taylor polynomial usually gives its best approximation near its center.

Accepted Answer

A Taylor polynomial is built so that its value and several derivatives match the original function at the center. Because of this matching, the polynomial has the same local behavior near the center, but the error often grows as x moves farther away.

Math: Taylor and Maclaurin Series

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