The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k + 1 } x ^ { k } } { k ^ { 2 } } = x - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 3 } } { 9 } - \cdots$ on its interval of convergence.
(a) Use the ratio test to determine the interval of convergence of the Maclaurin series for $f$. Show the work that leads to your answer.
(b) The Maclaurin series for $f$ evaluated at $x = \frac { 1 } { 4 }$ is an alternating series whose terms decrease in absolute value to 0. The approximation for $f \left( \frac { 1 } { 4 } \right)$ using the first two nonzero terms of this series is $\frac { 15 } { 64 }$. Show that this approximation differs from $f \left( \frac { 1 } { 4 } \right)$ by less than $\frac { 1 } { 500 }$.
(c) Let $h$ be the function defined by $h ( x ) = \int _ { 0 } ^ { x } f ( t ) \, d t$. Write the first three nonzero terms and the general term of the Maclaurin series for $h$.

Let $f$ be the function defined by $f ( x ) = e ^ { 2 x }$. Which of the following is the Maclaurin series for $f ^ { \prime }$, the derivative of $f$?
(A) $1 + x + \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 3 } } { 3 ! } + \cdots + \frac { x ^ { n } } { n ! } + \cdots$
(B) $2 + 2 x + \frac { 2 x ^ { 2 } } { 2 ! } + \frac { 2 x ^ { 3 } } { 3 ! } + \cdots + \frac { 2 x ^ { n } } { n ! } + \cdots$
(C) $1 + 2 x + \frac { ( 2 x ) ^ { 2 } } { 2 ! } + \frac { ( 2 x ) ^ { 3 } } { 3 ! } + \cdots + \frac { ( 2 x ) ^ { n } } { n ! } + \cdots$
(D) $2 + 2 ( 2 x ) + \frac { 2 ( 2 x ) ^ { 2 } } { 2 ! } + \frac { 2 ( 2 x ) ^ { 3 } } { 3 ! } + \cdots + \frac { 2 ( 2 x ) ^ { n } } { n ! } + \cdots$

The function $f$ has derivatives of all orders for all real numbers. It is known that $\left| f ^ { ( 4 ) } ( x ) \right| \leq \frac { 12 } { 5 }$ and $\left| f ^ { ( 5 ) } ( x ) \right| \leq \frac { 3 } { 2 }$ for $0 \leq x \leq 2$. Let $P _ { 4 } ( x )$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. The Taylor series for $f$ about $x = 0$ converges at $x = 2$. Of the following, which is the smallest value of $k$ for which the Lagrange error bound guarantees that $\left| f ( 2 ) - P _ { 4 } ( 2 ) \right| \leq k$?
(A) $\frac { 2 ^ { 5 } } { 5 ! } \cdot \frac { 3 } { 2 }$
(B) $\frac { 2 ^ { 5 } } { 5 ! } \cdot \frac { 12 } { 5 }$
(C) $\frac { 2 ^ { 4 } } { 4 ! } \cdot \frac { 3 } { 2 }$
(D) $\frac { 2 ^ { 4 } } { 4 ! } \cdot \frac { 12 } { 5 }$

Let $f$ be a function that has derivatives of all orders for all real numbers. Assume $f(0) = 5$, $f'(0) = -3$, $f''(0) = 1$, and $f'''(0) = 4$.
(a) Write the third-degree Taylor polynomial for $f$ about $x = 0$ and use it to approximate $f(0.2)$.
(b) Write the fourth-degree Taylor polynomial for $g$, where $g(x) = f\left(x^{2}\right)$, about $x = 0$.
(c) Write the third-degree Taylor polynomial for $h$, where $h(x) = \int_{0}^{x} f(t)\, dt$, about $x = 0$.
(d) Let $h$ be defined as in part (c). Given that $f(1) = 3$, either find the exact value of $h(1)$ or explain why it cannot be determined.

The function $f$ has derivatives of all orders for all real numbers $x$. Assume $f(2) = -3$, $f'(2) = 5$, $f''(2) = 3$, and $f'''(2) = -8$.
(a) Write the third-degree Taylor polynomial for $f$ about $x = 2$ and use it to approximate $f(1.5)$.
(b) The fourth derivative of $f$ satisfies the inequality $\left|f^{(4)}(x)\right| \leq 3$ for all $x$ in the closed interval $[1.5, 2]$. Use the Lagrange error bound on the approximation to $f(1.5)$ found in part (a) to explain why $f(1.5) \neq -5$.
(c) Write the fourth-degree Taylor polynomial, $P(x)$, for $g(x) = f\left(x^2 + 2\right)$ about $x = 0$. Use $P$ to explain why $g$ must have a relative minimum at $x = 0$.

The Maclaurin series for the function $f$ is given by $$f ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( 2 x ) ^ { n + 1 } } { n + 1 } = 2 x + \frac { 4 x ^ { 2 } } { 2 } + \frac { 8 x ^ { 3 } } { 3 } + \frac { 16 x ^ { 4 } } { 4 } + \cdots + \frac { ( 2 x ) ^ { n + 1 } } { n + 1 } + \cdots$$ on its interval of convergence.
(a) Find the interval of convergence of the Maclaurin series for $f$. Justify your answer.
(b) Find the first four terms and the general term for the Maclaurin series for $f ^ { \prime } ( x )$.
(c) Use the Maclaurin series you found in part (b) to find the value of $f ^ { \prime } \left( - \frac { 1 } { 3 } \right)$.

The Maclaurin series for $\ln\left(\dfrac{1}{1-x}\right)$ is $\displaystyle\sum_{n=1}^{\infty} \frac{x^n}{n}$ with interval of convergence $-1 \leq x < 1$.
(a) Find the Maclaurin series for $\ln\left(\dfrac{1}{1+3x}\right)$ and determine the interval of convergence.
(b) Find the value of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$.
(c) Give a value of $p$ such that $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n}{n^p}$ converges, but $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{2p}}$ diverges. Give reasons why your value of $p$ is correct.
(d) Give a value of $p$ such that $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^p}$ diverges, but $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{2p}}$ converges. Give reasons why your value of $p$ is correct.

The function $f$ has a Taylor series about $x = 2$ that converges to $f(x)$ for all $x$ in the interval of convergence. The $n$th derivative of $f$ at $x = 2$ is given by $f^{(n)}(2) = \frac{(n+1)!}{3^n}$ for $n \geq 1$, and $f(2) = 1$.
(a) Write the first four terms and the general term of the Taylor series for $f$ about $x = 2$.
(b) Find the radius of convergence for the Taylor series for $f$ about $x = 2$. Show the work that leads to your answer.
(c) Let $g$ be a function satisfying $g(2) = 3$ and $g'(x) = f(x)$ for all $x$. Write the first four terms and the general term of the Taylor series for $g$ about $x = 2$.
(d) Does the Taylor series for $g$ as defined in part (c) converge at $x = -2$? Give a reason for your answer.

Let $f$ be a function having derivatives of all orders for all real numbers. The third-degree Taylor polynomial for $f$ about $x = 2$ is given by $$T ( x ) = 7 - 9 ( x - 2 ) ^ { 2 } - 3 ( x - 2 ) ^ { 3 } .$$ (a) Find $f$ (2) and $f ^ { \prime \prime } ( 2 )$.
(b) Is there enough information given to determine whether $f$ has a critical point at $x = 2$ ? If not, explain why not. If so, determine whether $f ( 2 )$ is a relative maximum, a relative minimum, or neither, and justify your answer.
(c) Use $T ( x )$ to find an approximation for $f ( 0 )$. Is there enough information given to determine whether $f$ has a critical point at $x = 0$ ? If not, explain why not. If so, determine whether $f ( 0 )$ is a relative maximum, a relative minimum, or neither, and justify your answer.
(d) The fourth derivative of $f$ satisfies the inequality $\left| f ^ { ( 4 ) } ( x ) \right| \leq 6$ for all $x$ in the closed interval $[ 0,2 ]$. Use the Lagrange error bound on the approximation to $f ( 0 )$ found in part (c) to explain why $f ( 0 )$ is negative.

Let $f$ be a function with derivatives of all orders and for which $f ( 2 ) = 7$. When $n$ is odd, the $n$th derivative of $f$ at $x = 2$ is 0. When $n$ is even and $n \geq 2$, the $n$th derivative of $f$ at $x = 2$ is given by $f ^ { ( n ) } ( 2 ) = \frac { ( n - 1 ) ! } { 3 ^ { n } }$.
(a) Write the sixth-degree Taylor polynomial for $f$ about $x = 2$.
(b) In the Taylor series for $f$ about $x = 2$, what is the coefficient of $( x - 2 ) ^ { 2 n }$ for $n \geq 1$ ?
(c) Find the interval of convergence of the Taylor series for $f$ about $x = 2$. Show the work that leads to your answer.

The function $f$ is defined by the power series $$f(x) = -\frac{x}{2} + \frac{2x^{2}}{3} - \frac{3x^{3}}{4} + \cdots + \frac{(-1)^{n} n x^{n}}{n+1} + \cdots$$ for all real numbers $x$ for which the series converges. The function $g$ is defined by the power series $$g(x) = 1 - \frac{x}{2!} + \frac{x^{2}}{4!} - \frac{x^{3}}{6!} + \cdots + \frac{(-1)^{n} x^{n}}{(2n)!} + \cdots$$ for all real numbers $x$ for which the series converges.
(a) Find the interval of convergence of the power series for $f$. Justify your answer.
(b) The graph of $y = f(x) - g(x)$ passes through the point $(0, -1)$. Find $y'(0)$ and $y''(0)$. Determine whether $y$ has a relative minimum, a relative maximum, or neither at $x = 0$. Give a reason for your answer.

Let $f$ be the function given by $f(x) = e^{-x^2}$.
(a) Write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 0$.
(b) Use your answer to part (a) to find $\lim_{x \to 0} \frac{1 - x^2 - f(x)}{x^4}$.
(c) Write the first four nonzero terms of the Taylor series for $\int_{0}^{x} e^{-t^2}\, dt$ about $x = 0$. Use the first two terms of your answer to estimate $\int_{0}^{1/2} e^{-t^2}\, dt$.
(d) Explain why the estimate found in part (c) differs from the actual value of $\int_{0}^{1/2} e^{-t^2}\, dt$ by less than $\frac{1}{200}$.

Let $h$ be a function having derivatives of all orders for $x > 0$. Selected values of $h$ and its first four derivatives are indicated in the table below. The function $h$ and these four derivatives are increasing on the interval $1 \leq x \leq 3$.

$x$	$h ( x )$	$h ^ { \prime } ( x )$	$h ^ { \prime \prime } ( x )$	$h ^ { \prime \prime \prime } ( x )$	$h ^ { ( 4 ) } ( x )$
1	11	30	42	99	18
2	80	128	$\frac { 488 } { 3 }$	$\frac { 448 } { 3 }$	$\frac { 584 } { 9 }$
3	317	$\frac { 753 } { 2 }$	$\frac { 1383 } { 4 }$	$\frac { 3483 } { 16 }$	$\frac { 1125 } { 16 }$

(a) Write the first-degree Taylor polynomial for $h$ about $x = 2$ and use it to approximate $h ( 1.9 )$. Is this approximation greater than or less than $h ( 1.9 )$ ? Explain your reasoning.
(b) Write the third-degree Taylor polynomial for $h$ about $x = 2$ and use it to approximate $h ( 1.9 )$.
(c) Use the Lagrange error bound to show that the third-degree Taylor polynomial for $h$ about $x = 2$ approximates $h ( 1.9 )$ with error less than $3 \times 10 ^ { - 4 }$.

The Maclaurin series for $e^{x}$ is $e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \cdots + \frac{x^{n}}{n!} + \cdots$. The continuous function $f$ is defined by $f(x) = \frac{e^{(x-1)^{2}} - 1}{(x-1)^{2}}$ for $x \neq 1$ and $f(1) = 1$. The function $f$ has derivatives of all orders at $x = 1$.
(a) Write the first four nonzero terms and the general term of the Taylor series for $e^{(x-1)^{2}}$ about $x = 1$.
(b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 1$.
(c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b).
(d) Use the Taylor series for $f$ about $x = 1$ to determine whether the graph of $f$ has any points of inflection.

The function $f$, defined by $$f(x) = \begin{cases} \frac{\cos x - 1}{x^2} & \text{for } x \neq 0 \\ -\frac{1}{2} & \text{for } x = 0 \end{cases}$$ has derivatives of all orders. Let $g$ be the function defined by $g(x) = 1 + \int_{0}^{x} f(t)\,dt$.
(a) Write the first three nonzero terms and the general term of the Taylor series for $\cos x$ about $x = 0$. Use this series to write the first three nonzero terms and the general term of the Taylor series for $f$ about $x = 0$.
(b) Use the Taylor series for $f$ about $x = 0$ found in part (a) to determine whether $f$ has a relative maximum, relative minimum, or neither at $x = 0$. Give a reason for your answer.
(c) Write the fifth-degree Taylor polynomial for $g$ about $x = 0$.
(d) The Taylor series for $g$ about $x = 0$, evaluated at $x = 1$, is an alternating series with individual terms that decrease in absolute value to 0. Use the third-degree Taylor polynomial for $g$ about $x = 0$ to estimate the value of $g(1)$. Explain why this estimate differs from the actual value of $g(1)$ by less than $\frac{1}{6!}$.

Let $f(x) = \sin\left(x^2\right) + \cos x$. The graph of $y = \left|f^{(5)}(x)\right|$ is shown above.
(a) Write the first four nonzero terms of the Taylor series for $\sin x$ about $x = 0$, and write the first four nonzero terms of the Taylor series for $\sin\left(x^2\right)$ about $x = 0$.
(b) Write the first four nonzero terms of the Taylor series for $\cos x$ about $x = 0$. Use this series and the series for $\sin\left(x^2\right)$, found in part (a), to write the first four nonzero terms of the Taylor series for $f$ about $x = 0$.
(c) Find the value of $f^{(6)}(0)$.
(d) Let $P_4(x)$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. Using information from the graph of $y = \left|f^{(5)}(x)\right|$ shown above, show that $\left|P_4\left(\frac{1}{4}\right) - f\left(\frac{1}{4}\right)\right| < \frac{1}{3000}$.

The function $f$ is twice differentiable for $x > 0$ with $f(1) = 15$ and $f''(1) = 20$. Values of $f'$, the derivative of $f$, are given for selected values of $x$ in the table below.

$x$	1	1.1	1.2	1.3	1.4
$f'(x)$	8	10	12	13	14.5

(a) Write an equation for the line tangent to the graph of $f$ at $x = 1$. Use this line to approximate $f(1.4)$.
(b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate $\int_{1}^{1.4} f'(x)\, dx$. Use the approximation for $\int_{1}^{1.4} f'(x)\, dx$ to estimate the value of $f(1.4)$. Show the computations that lead to your answer.
(c) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f(1.4)$. Show the computations that lead to your answer.
(d) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f(1.4)$.

The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( - \frac { x } { 4 } \right) ^ { n }$. What is the value of $f ( 3 )$ ?
(A) - 3
(B) $- \frac { 3 } { 7 }$
(C) $\frac { 4 } { 7 }$
(D) $\frac { 13 } { 16 }$
(E) 4

The function $g$ has derivatives of all orders, and the Maclaurin series for $g$ is
$$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n + 1 } } { 2 n + 3 } = \frac { x } { 3 } - \frac { x ^ { 3 } } { 5 } + \frac { x ^ { 5 } } { 7 } - \cdots$$
(a) Using the ratio test, determine the interval of convergence of the Maclaurin series for $g$.
(b) The Maclaurin series for $g$ evaluated at $x = \frac { 1 } { 2 }$ is an alternating series whose terms decrease in absolute value to 0. The approximation for $g \left( \frac { 1 } { 2 } \right)$ using the first two nonzero terms of this series is $\frac { 17 } { 120 }$. Show that this approximation differs from $g \left( \frac { 1 } { 2 } \right)$ by less than $\frac { 1 } { 200 }$.
(c) Write the first three nonzero terms and the general term of the Maclaurin series for $g ^ { \prime } ( x )$.

For $x > 0$, the power series $1 - \frac { x ^ { 2 } } { 3 ! } + \frac { x ^ { 4 } } { 5 ! } - \frac { x ^ { 6 } } { 7 ! } + \cdots + ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { ( 2 n + 1 ) ! } + \cdots$ converges to which of the following?
(A) $\cos x$
(B) $\sin x$
(C) $\frac { \sin x } { x }$
(D) $e ^ { x } - e ^ { x ^ { 2 } }$
(E) $1 + e ^ { x } - e ^ { x ^ { 2 } }$

Let $f$ be a function having derivatives of all orders for $x > 0$ such that $f ( 3 ) = 2 , f ^ { \prime } ( 3 ) = - 1 , f ^ { \prime \prime } ( 3 ) = 6$, and $f ^ { \prime \prime \prime } ( 3 ) = 12$. Which of the following is the third-degree Taylor polynomial for $f$ about $x = 3$ ?
(A) $2 - x + 6 x ^ { 2 } + 12 x ^ { 3 }$
(B) $2 - x + 3 x ^ { 2 } + 2 x ^ { 3 }$
(C) $2 - ( x - 3 ) + 6 ( x - 3 ) ^ { 2 } + 12 ( x - 3 ) ^ { 3 }$
(D) $2 - ( x - 3 ) + 3 ( x - 3 ) ^ { 2 } + 4 ( x - 3 ) ^ { 3 }$
(E) $2 - ( x - 3 ) + 3 ( x - 3 ) ^ { 2 } + 2 ( x - 3 ) ^ { 3 }$

A function $f$ has derivatives of all orders at $x = 0$. Let $P _ { n } ( x )$ denote the $n$th-degree Taylor polynomial for $f$ about $x = 0$.
(a) It is known that $f ( 0 ) = - 4$ and that $P _ { 1 } \left( \frac { 1 } { 2 } \right) = - 3$. Show that $f ^ { \prime } ( 0 ) = 2$.
(b) It is known that $f ^ { \prime \prime } ( 0 ) = - \frac { 2 } { 3 }$ and $f ^ { \prime \prime \prime } ( 0 ) = \frac { 1 } { 3 }$. Find $P _ { 3 } ( x )$.
(c) The function $h$ has first derivative given by $h ^ { \prime } ( x ) = f ( 2 x )$. It is known that $h ( 0 ) = 7$. Find the third-degree Taylor polynomial for $h$ about $x = 0$.

A function $f$ has derivatives of all orders for $-1 < x < 1$. The derivatives of $f$ satisfy the conditions below. The Maclaurin series for $f$ converges to $f(x)$ for $|x| < 1$. $$\begin{aligned} f(0) &= 0 \\ f'(0) &= 1 \\ f^{(n+1)}(0) &= -n \cdot f^{(n)}(0) \text{ for all } n \geq 1 \end{aligned}$$ (a) 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 $f$.
(b) Determine whether the Maclaurin series described in part (a) converges absolutely, converges conditionally, or diverges at $x = 1$. Explain your reasoning.
(c) Write the first four nonzero terms and the general term of the Maclaurin series for $g(x) = \int_{0}^{x} f(t)\, dt$.
(d) 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 $\left|P_4\!\left(\frac{1}{2}\right) - g\!\left(\frac{1}{2}\right)\right| < \frac{1}{500}$.

The Maclaurin series for $\ln ( 1 + x )$ 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 $\ln ( 1 + x )$. Let $f$ be the function defined by
$$f ( x ) = x \ln \left( 1 + \frac { x } { 3 } \right)$$
(a) Write the first four nonzero terms and the general term of the Maclaurin series for $f$.
(b) Determine the interval of convergence of the Maclaurin series for $f$. Show the work that leads to your answer.
(c) Let $P _ { 4 } ( x )$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. Use the alternating series error bound to find an upper bound for $\left| P _ { 4 } ( 2 ) - f ( 2 ) \right|$.

A function $f$ has derivatives of all orders for all real numbers $x$. A portion of the graph of $f$ is shown above, 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.

$n$	$f ^ { ( n ) } ( 0 )$
2	3
3	$-\frac { 23 } { 2 }$
4	54

(a) Write the third-degree Taylor polynomial for $f$ about $x = 0$.
(b) Write the first three nonzero terms of the Maclaurin series for $e ^ { x }$. Write the second-degree Taylor polynomial for $e ^ { x } f ( x )$ about $x = 0$.
(c) Let $h$ be the function defined by $h ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$. Use the Taylor polynomial found in part (a) to find an approximation for $h ( 1 )$.
(d) It is known that the Maclaurin series for $h$ converges to $h ( x )$ for all real numbers $x$. It is also known that the individual terms of the series for $h ( 1 )$ 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 $h ( 1 )$ by at most 0.45.