Practice building, using, and interpreting Taylor and Maclaurin series for common functions.
Read each problem carefully. Show your work in the space provided. Use factorial notation when helpful, and include enough terms to make your reasoning clear.
Approximating functions with polynomials
Math - Grade 9-12
- 1
Write the first four nonzero terms of the Maclaurin series for e^x.
- 2
Write the first four nonzero terms of the Maclaurin series for sin x.
- 3
Write the first four nonzero terms of the Maclaurin series for cos x.
- 4
Use the Maclaurin polynomial 1 + x + x^2/2 to approximate e^0.2. Round your answer to four decimal places.
- 5
Use the Maclaurin polynomial x - x^3/6 to approximate sin(0.3). Round your answer to four decimal places.
- 6
Find the third-degree Taylor polynomial for f(x) = ln x centered at x = 1.
- 7
Find the second-degree Taylor polynomial for f(x) = sqrt(x) centered at x = 4.
- 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).
- 9
For the geometric series 1 + x + x^2 + x^3 + ... , state the interval of convergence.
- 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.
- 11
Find the coefficient of x^5 in the Maclaurin series for sin x.
- 12
Explain why a Taylor polynomial usually gives its best approximation near its center.