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

The function $f$ has derivatives of all orders for all real numbers $x$. A portion of the graph of $f$ is shown above, including the tangent line to the graph of $f$ at $x=0$. Selected derivatives for $f$ at $x=0$ are given in the table below.

Write the third-degree Taylor polynomial for $f$ about $x=0$.

Write the first three nonzero terms of the Taylor series for $e^x$ about $x=0$. Write the second-degree Taylor polynomial for $e^{x}f\left(x\right)$ about $x=0$.

The function $f$ has derivatives of all orders for all real numbers. It is known that $f\left(0\right)=4$ and $f\text{'}\left(0\right)=-3$.

Let $g$ be the function such that $g\left(0\right)=5$ and $g^{\text{'}}\left(x\right)=e^{2x}·f\left(x\right)$. Write the second-degree Taylor polynomial for $g$ about $x=0$.

The Taylor series for $\ln \left(1+x\right)$ about $x=0$ is given by

$x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+...+\frac{{\left(-1\right)}^{n+1}{x}^n}{n}+...$,

which converges to $\ln \left(1+x\right)$ on its interval of convergence.

Let $f$ be the function defined by $f\left(x\right)=x^2\ln \left(1+\frac{x}{2}\right)$.

Write the first four nonzero terms and the general term of the Taylor series for $f$ about $x=0$.

Determine the interval of convergence of the Taylor series for $f$ about $x=0$. Show the work that leads to your answer.

The Taylor series for a function $f$ about $x=2$ is given by $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{3^n}{2n}{\left(x-2\right)}^n$. Find the first three nonzero terms and the general term of the Taylor series for $f\text{'}$, the derivative of $f$, about $x=2$.

The Maclaurin series for a function $f$ is given by $f\left(x\right)=\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n{\left(2x\right)}^{2n}}{\left(2n\right)!}=1-2x^2+\frac{2x^4}{3}-\frac{4x^6}{45}+...$, which converges for all real numbers $x$. If the first three nonzero terms of the series are used to approximate $f\left(x\right)$ on the interval $-\frac{1}{2}\le x\le \frac{1}{3}$, use the alternating series error bound to determine an upper bound for the error of the approximations.

A function $f$ has derivatives of all orders for all real numbers $x$. The fourth-degree Taylor polynomial for $f$ about $x=2$ is used to approximate $f\left(1.9\right)$. Given that $\left|f^{\left(5\right)}\left(x\right)\right|\le 9$ for $1.5\le x\le 2$, use the Lagrange error bound to show that this approximation is within $\frac{1}{{10}^6}$ of the exact value of $f\left(1.9\right)$.

Let $y=f\left(x\right)$ be the particular solution to the differential equation $\frac{\text{d}y}{\text{d}x}=y·\left(x\text{ln}x\right)$ with initial condition $f\left(1\right)=4$. It can be shown that $f\text{'}\text{'}\left(1\right)=4$.

Write the second-degree Taylor polynomial for $f$ about $x=1$. Use the Taylor polynomial to approximate $f\left(2\right)$.

It can be shown that $f(-3)=\sum _{n=1}^{\infty }\frac{(n+1)(-3{)}^n}{n^2{6}^n}=\sum _{n=1}^{\infty }\frac{n+1}{n^2}{\left(-\frac{1}{2}\right)}^n$ and that the first three terms of this series sum to $S_3=-\frac{125}{144}$. Show that $\left|f(-3)-S_3\right|<\frac{1}{50}$.

Let $g\left(x\right)=\sum _{n=1}^{\infty }\frac{(n+1)x^{2n}}{n^2{3}^n}$. Use the ratio test to determine the radius of convergence of the Maclaurin series for $g$.

Let $y=f\left(x\right)$ be the particular solution to the differential equation $\frac{dy}{dx}=2x^{2}y·\ln x$ with the initial condition $f\left(1\right)=5$. It can be shown that $f^{\text{'}\text{'}}\left(1\right)=10$.

Write the second-degree Taylor polynomial for $f$ about $x=1$, and use the polynomial to approximate $f\left(2\right)$.

The function $f$ has derivatives of all orders for all real numbers. It is known that $f\left(0\right)=3$, $f\text{'}\left(0\right)=-2$, and $f^{\text{'}\text{'}}\left(x\right)=f\left(2x^2\right)$. Write the fourth-degree Taylor polynomial for $f$ about $x=0$. Show the work that leads to your answer.

A function $f$ is such that $f\left(3\right)=2$ and the Taylor series of its derivative $f^{\text{'}}$ is given by $f^{\text{'}}\left(x\right)=\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{5}^n{\left(x-3\right)}^{n-1}$. Use this function to determine $f$ explicitly within the radius of convergence of the series.

The Maclaurin series for a function $f$ is given by $f\left(x\right)=x^2-\frac{x^4}{2}+\frac{x^6}{3}-\frac{x^8}{4}+...+{\left(-1\right)}^{n+1}·\frac{x^{2n}}{n}+...$, which converges on $\left[-1,1\right]$.

Write the first four nonzero terms of the Maclaurin series for $f^{\text{'}}\left(t^4\right)$. Given that $g\left(x\right)={\int }_0^x{f}^{\text{'}}\left(t^4\right)dt$, use the first two nonzero terms of the Maclaurin series for $g$ to approximate $g\left(1\right)$.

Show that your approximation in part (a) must differ from $g\left(1\right)$ by less than $\frac{1}{10}$. Justify your answer.

Let the function $f$ be defined by $f\left(x\right)=\ln \left(1+2x^2\right)$. The Taylor series for $f$ about $x=0$ is given by $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{2^n{x}^{2n}}{n}$, which converges on the interval $\left[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right]$. Use the Taylor series to find a rational number $A$ such that $\left|A-\ln \left(\frac{3}{2}\right)\right|<\frac{1}{50}$. Justify your answer.

Let the function $f$ be defined by $f\left(x\right)=\sin \left(3x\right)$. The fourth-degree Taylor polynomial for $f$ about $x=1$ is used to approximate $f\left(x\right)$. Use the Lagrange error bound to show that this approximation will be within $\frac{7}{{10}^4}$ of the exact value of $f\left(x\right)$ for all $x$ in the interval $0.8\le x\le 1.2$.

For a function $f$, the Maclaurin series is given by $f\left(x\right)=\sum _{n=4}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n-1}·{\left(\frac{x}{3}\right)}^n$. For $\left|x\right|<R$, where $R$ is the radius of convergence of the series, show that $y=f\left(x\right)$ is a solution to the differential equation $y-xy\text{'}=\frac{x^4}{27\left(x+3\right)}$.

The function $f$ has a Taylor series about $x=1$ that converges to $f\left(x\right)$ for all $x$ in the interval of convergence. It is known that $f\left(1\right)=1$, $f\text{'}\left(1\right)=-\frac{1}{2}$, and the $n$th derivative of $f$ at $x=1$ is given by $f^{\left(n\right)}\left(1\right)=(-1{)}^n\frac{(n-1)!}{2^n}$ for $n\ge 2$.

Write the first four nonzero terms and the general term of the Taylor series for $f$ about $x=1$.

The Taylor series for $f$ about $x=1$ has a radius of convergence of $2$. Find the interval of convergence. Show the work that leads to your answer.

The Taylor series for $f$ about $x=1$ can be used to represent $f(1.2)$ as an alternating series. Use the first three nonzero terms of the alternating series to approximate $f(1.2)$.

Show that the approximation found in part (c) is within $0.001$ of the exact value of $f(1.2)$.

The function $f$ is defined by the power series $f\left(x\right)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots +\frac{(-1{)}^n{x}^{2n+1}}{2n+1}+\cdots$ for all real numbers $x$ for which the series converges.

Using the ratio test, find the interval of convergence of the power series for $f$. Justify your answer.

Show that $\left|f\left(\frac{1}{2}\right)-\frac{1}{2}\right|<\frac{1}{10}$. Justify your answer.

Write the first four nonzero terms and the general term for an infinite series that represents $f\text{'}\left(x\right)$.

The Maclaurin series for $\text{ln}(1+x)$ is given by

$x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots +(-1{)}^{n+1}\frac{x^n}{n}+\cdots$

On its interval of convergence, this series converges to $\text{ln}(1+x)$. Let $f$ be the function defined by $f\left(x\right)=x\text{ln}\left(1+\frac{x}{3}\right)$.

Write the first four nonzero terms and the general term of the Maclaurin series for $f$.

Determine the interval of convergence of the Maclaurin series for $f$. Show the work that leads to your answer.

Let $P_4\left(x\right)$ be the fourth-degree Taylor polynomial for $f$ about $x=0$. Use the alternating series error bound to find an upper bound for $\left|P_4\left(2\right)-f\left(2\right)\right|$.

The Taylor series for a function $f$ about $x=4$ is given by

$\sum _{n=1}^{\infty }\frac{(x-4{)}^{n+1}}{(n+1)3^n}=\frac{(x-4{)}^2}{2·3}+\frac{(x-4{)}^3}{3·3^2}+\frac{(x-4{)}^4}{4·3^3}+\cdots +\frac{(x-4{)}^{n+1}}{(n+1)3^n}+\cdots$

and converges to $f\left(x\right)$ on its interval of convergence.

Using the ratio test, find the interval of convergence of the Taylor series for $f$ about $x=4$. Justify your answer.

Find the first three nonzero terms and the general term of the Taylor series for $f\text{'}$, the derivative of $f$, about $x=4$.

A function $f$ has derivatives of all orders for $-1<x<1$. The derivatives of $f$ satisfy the following.

$f\left(0\right)=0$

$f\text{'}\left(0\right)=1$

$f^{(n+1)}\left(0\right)=-n{f}^{\left(n\right)}\left(0\right)$ for all $n\ge 1$

The Maclaurin series for $f$ converges to $f\left(x\right)$ for $\left|x\right|<1$.

Show that the first four nonzero terms of the Maclaurin series for $f$ are $x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}$, and write the general term of the Maclaurin series for $f$.

A function $f$ has derivatives of all orders for all real numbers $x$. A portion of the graph of $f$ is shown below, along with the line tangent to the graph of $f$ at $x=0$. Selected derivatives of $f$ at $x=0$ are given in the table below.

Write the third-degree Taylor polynomial for $f$ about $x=0$.

Write the first three nonzero terms of the Maclaurin series for $e^x$. Write the second-degree Taylor polynomial for $e^{x}f\left(x\right)$ about $x=0$.

Let $h$ be the function defined by $h\left(x\right)={\int }_0^{x}f\left(t\right)\text{d}t$. Use the Taylor polynomial found in part (a) to find an approximation for $h\left(1\right)$.

It is known that the Maclaurin series for $h$ converges to $h\left(x\right)$ for all real numbers $x$. It is also known that the individual terms of the series for $h\left(1\right)$ alternate in sign and decrease in absolute value to $0$. Use the alternating series error bound to show that the approximation found in part (c) differs from $h\left(1\right)$ by at most $0.45$.

The function $f$ has derivatives of all orders for all real numbers. It is known that $f\left(0\right)=2$, $f\text{'}\left(0\right)=3$, $f\text{'}\text{'}\left(x\right)=-f\left(x^2\right)$, and $f\text{'}\text{'}\text{'}\left(x\right)=-2x·f\text{'}\left(x^2\right)$.

Find $f^{\left(4\right)}\left(x\right)$, the fourth derivative of $f$ with respect to $x$. Write the fourth-degree Taylor polynomial for $f$ about $x=0$. Show the work that leads to your answer.

The fourth-degree Taylor polynomial for $f$ about $x=0$ is used to approximate $f(0.1)$. Given that $\left|f^{\left(5\right)}\left(x\right)\right|\le 15$ for $0\le x\le 0.5$, use the Lagrange error bound to show that this approximation is within $\frac{1}{{10}^5}$ of the exact value of $f(0.1)$.

Let $g$ be the function such that $g\left(0\right)=4$ and $g\text{'}\left(x\right)=e^{x}f\left(x\right)$. Write the second-degree Taylor polynomial for $g$ about $x=0$.