Series Representations of Functions (College Board AP® Calculus BC): Exam Questions

The function has derivatives of all orders for all real numbers . A portion of the graph of is shown above, including the tangent line to the graph of at . Selected derivatives for at are given in the table below.

Write the third-degree Taylor polynomial for about .

Write the first three nonzero terms of the Taylor series for about . Write the second-degree Taylor polynomial for about .

The function has derivatives of all orders for all real numbers. It is known that and .

Let be the function such that and . Write the second-degree Taylor polynomial for about .

The Taylor series for about is given by

,

which converges to on its interval of convergence.

Let be the function defined by .

Write the first four nonzero terms and the general term of the Taylor series for about .

Determine the interval of convergence of the Taylor series for about . Show the work that leads to your answer.

The Taylor series for a function about is given by . Find the first three nonzero terms and the general term of the Taylor series for , the derivative of , about .

The Maclaurin series for a function is given by , which converges for all real numbers . If the first three nonzero terms of the series are used to approximate on the interval , use the alternating series error bound to determine an upper bound for the error of the approximations.

A function has derivatives of all orders for all real numbers . The fourth-degree Taylor polynomial for about is used to approximate . Given that for , use the Lagrange error bound to show that this approximation is within of the exact value of .

Let be the particular solution to the differential equation with the initial condition . It can be shown that .

Write the second-degree Taylor polynomial for about , and use the polynomial to approximate .

The function has derivatives of all orders for all real numbers. It is known that , , and . Write the fourth-degree Taylor polynomial for about . Show the work that leads to your answer.

A function is such that and the Taylor series of its derivative is given by . Use this function to determine explicitly within the radius of convergence of the series.

The Maclaurin series for a function is given by , which converges on .

Write the first four nonzero terms of the Maclaurin series for . Given that , use the first two nonzero terms of the Maclaurin series for to approximate .

Show that your approximation in part (a) must differ from by less than . Justify your answer.

Let the function be defined by . The Taylor series for about is given by , which converges on the interval . Use the Taylor series to find a rational number such that . Justify your answer.

Let the function be defined by . The fourth-degree Taylor polynomial for about is used to approximate . Use the Lagrange error bound to show that this approximation will be within of the exact value of for all in the interval .

For a function , the Maclaurin series is given by . For , where is the radius of convergence of the series, show that is a solution to the differential equation .