Unit 10 Overview (College Board AP® Calculus BC): Exam Questions

Let $y=f\left(x\right)$ be the particular solution to the differential equation $\frac{\text{d}y}{\text{d}x}=(3-x)y^2$ with initial condition $f\left(1\right)=-1$.

Find $f\text{'}\text{'}\left(1\right)$, the value of $\frac{{\text{d}}^{2}y}{\text{d}{x}^2}$ at the point $(1,-1)$. Show the work that leads to your answer.

Write the second-degree Taylor polynomial for $f$ about $x=1$.

The second-degree Taylor polynomial for $f$ about $x=1$ is used to approximate $f(1.1)$. Given that $\left|f\text{'}\text{'}\text{'}\left(x\right)\right|\le 60$ for all $x$ in the interval $1\le x\le 1.1$, use the Lagrange error bound to show that this approximation differs from $f(1.1)$ by at most $0.01$.

Use Euler's method, starting at $x=1$ with two steps of equal size, to approximate $f(1.4)$. Show the work that leads to your answer.

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 $f\text{'}\left(x\right)$.

Use the result from part (c) to find the value of $f\text{'}\left(\frac{1}{6}\right)$.

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 $x=4$.

The Taylor series for $f\text{'}$ described in part (b) is a geometric series. For all $x$ in the interval of convergence of the Taylor series for $f\text{'}$, show that $f\text{'}\left(x\right)=\frac{x-4}{7-x}$.

It is known that the radius of convergence of the Taylor series for $f$ about $x=4$ is the same as the radius of convergence of the Taylor series for $f\text{'}$ about $x=4$. Does the Taylor series for $f\text{'}$ described in part (b) converge to $f\text{'}\left(x\right)=\frac{x-4}{7-x}$ at $x=8$? Give a reason for your answer.

The function $g$ has derivatives of all orders for all real numbers. The Maclaurin series for $g$ is given by $g\left(x\right)=\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^n}{2e^n+3}$ on its interval of convergence.

State the conditions necessary to use the integral test to determine convergence of the series $\sum _{n=0}^{\infty }\frac{1}{e^n}$. Use the integral test to show that $\sum _{n=0}^{\infty }\frac{1}{e^n}$ converges.

Use the limit comparison test with the series $\sum _{n=0}^{\infty }\frac{1}{e^n}$ to show that the series $g\left(1\right)=\sum _{n=0}^{\infty }\frac{(-1{)}^n}{2e^n+3}$ converges absolutely.

Determine the radius of convergence of the Maclaurin series for $g$.

The first two terms of the series $g\left(1\right)=\sum _{n=0}^{\infty }\frac{(-1{)}^n}{2e^n+3}$ are used to approximate $g\left(1\right)$. Use the alternating series error bound to determine an upper bound on the error of the approximation.

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 $f$.

Determine whether the Maclaurin series described in part (a) converges absolutely, converges conditionally, or diverges at $x=1$. Explain your reasoning.

Write the first four nonzero terms and the general term of the Maclaurin series for $g\left(x\right)={\int }_0^{x}f\left(t\right)\text{d}t$.

Let $P_n\left(\frac{1}{2}\right)$ represent the $n$th-degree Taylor polynomial for $g$ about $x=0$ evaluated at $x=\frac{1}{2}$, where $g$ is the function defined in part (c). Use the alternating series error bound to show that $\left|P_4\left(\frac{1}{2}\right)-g\left(\frac{1}{2}\right)\right|<\frac{1}{500}$.

Let $f$ be the function defined by $f\left(x\right)=\frac{3}{2x^2-7x+5}$.

Find the slope of the line tangent to the graph of $f$ at $x=3$.

Find the $x$-coordinate of each critical point of $f$ in the interval $1<x<2.5$. Classify each critical point as the location of a relative minimum, a relative maximum, or neither. Justify your answers.

Using the identity that $\frac{3}{2x^2-7x+5}=\frac{2}{2x-5}-\frac{1}{x-1}$, evaluate ${\int }_5^{\infty }f\left(x\right)\text{d}x$ or show that the integral diverges.

Determine whether the series $\sum _{n=5}^{\infty }\frac{3}{2n^2-7n+5}$ converges or diverges. State the conditions of the test used for determining convergence or divergence.

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 $x=0$.

The Maclaurin series for a function $f$ is given by $\sum _{n=1}^{\infty }\frac{(n+1)x^n}{n^2{6}^n}$ and converges to $f\left(x\right)$ for all $x$ in the interval of convergence. It can be shown that the Maclaurin series for $f$ has a radius of convergence of $6$.

Determine whether the Maclaurin series for $f$ converges or diverges at $x=6$. Give a reason for your answer.

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 $\left|f(-3)-S_3\right|<\frac{1}{50}$.

Find the general term of the Maclaurin series for $f\text{'}$, the derivative of $f$. Find the radius of convergence of the Maclaurin series for $f\text{'}$.

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 $g$.