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 $h$.

13. $\sin ( 2 x ) =$
(A) $\quad x - \frac { x ^ { 3 } } { 3 ! } + \frac { x ^ { 5 } } { 5 ! } - \ldots + \frac { ( - 1 ) ^ { n - 1 } x ^ { 2 n - 1 } } { ( 2 n - 1 ) ! } + \ldots$
(B) $\quad 2 x - \frac { ( 2 x ) ^ { 3 } } { 3 ! } + \frac { ( 2 x ) ^ { 5 } } { 5 ! } - \ldots + \frac { ( - 1 ) ^ { n - 1 } ( 2 x ) ^ { 2 n - 1 } } { ( 2 n - 1 ) ! } + \ldots$
(C) $- \frac { ( 2 x ) ^ { 2 } } { 2 ! } + \frac { ( 2 x ) ^ { 4 } } { 4 ! } - \ldots + \frac { ( - 1 ) ^ { n } ( 2 x ) ^ { 2 n } } { ( 2 n ) ! } + \ldots$
(D) $\frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } + \frac { x ^ { 6 } } { 6 ! } + \ldots + \frac { x ^ { 2 n } } { ( 2 n ) ! } + \ldots$
(E) $\quad 2 x + \frac { ( 2 x ) ^ { 3 } } { 3 ! } + \frac { ( 2 x ) ^ { 5 } } { 5 ! } + \ldots + \frac { ( 2 x ) ^ { 2 n - 1 } } { ( 2 n - 1 ) ! } + \ldots$

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 $x = 0$.

3. The Taylor series about $x = 5$ for a certain function $f$ converges to $f ( x )$ for all $x$ in the interval of convergence. The $n$th derivative of $f$ at $x = 5$ is given by $f ^ { ( n ) } ( 5 ) = \frac { ( - 1 ) ^ { n } n ! } { 2 ^ { n } ( n + 2 ) }$, and $f ( 5 ) = \frac { 1 } { 2 }$.
(a) Write the third-degree Taylor polynomial for $f$ about $x = 5$.
(b) Find the radius of convergence of the Taylor series for $f$ about $x = 5$.
(c) Show that the sixth-degree Taylor polynomial for $f$ about $x = 5$ approximates $f ( 6 )$ with error less than $\frac { 1 } { 1000 }$.

END OF PART A OF SECTION II

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CALCULUS BC SECTION II, Part B Time-45 minutes Number of problems-3 No calculator is allowed for these problems.

A function $f$ is defined by $$f(x) = \frac{1}{3} + \frac{2}{3^2}x + \frac{3}{3^3}x^2 + \cdots + \frac{n+1}{3^{n+1}}x^n + \cdots$$ for all $x$ in the interval of convergence of the given power series.
(a) Find the interval of convergence for this power series. Show the work that leads to your answer.
(b) Find $\displaystyle\lim_{x \rightarrow 0} \frac{f(x) - \frac{1}{3}}{x}$.
(c) Write the first three nonzero terms and the general term for an infinite series that represents $\displaystyle\int_0^1 f(x)\, dx$.
(d) Find the sum of the series determined in part (c).

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.

6. The function $f$ is defined by the power series
$$f ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { 2 n } } { ( 2 n + 1 ) ! } = 1 - \frac { x ^ { 2 } } { 3 ! } + \frac { x ^ { 4 } } { 5 ! } - \frac { x ^ { 6 } } { 7 ! } + \cdots + \frac { ( - 1 ) ^ { n } x ^ { 2 n } } { ( 2 n + 1 ) ! } + \cdots$$
for all real numbers $x$.
(a) Find $f ^ { \prime } ( 0 )$ and $f ^ { \prime \prime } ( 0 )$. Determine whether $f$ has a local maximum, a local minimum, or neither at $x = 0$. Give a reason for your answer.
(b) Show that $1 - \frac { 1 } { 3 ! }$ approximates $f ( 1 )$ with error less than $\frac { 1 } { 100 }$.
(c) Show that $y = f ( x )$ is a solution to the differential equation $x y ^ { \prime } + y = \cos x$.

END OF EXAMINATION

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 the function given by $f ( x ) = \sin \left( 5 x + \frac { \pi } { 4 } \right)$, and let $P ( x )$ be the third-degree Taylor polynomial for $f$ about $x = 0$.
(a) Find $P ( x )$.
(b) Find the coefficient of $x ^ { 22 }$ in the Taylor series for $f$ about $x = 0$.
(c) Use the Lagrange error bound to show that $\left| f \left( \frac { 1 } { 10 } \right) - P \left( \frac { 1 } { 10 } \right) \right| < \frac { 1 } { 100 }$.
(d) Let $G$ be the function given by $G ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$. Write the third-degree Taylor polynomial for $G$ about $x = 0$.

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.

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 $\left| P _ { 4 } \left( \frac { 1 } { 4 } \right) - f \left( \frac { 1 } { 4 } \right) \right| < \frac { 1 } { 3000 }$.

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 $f$ be the function given by $f ( x ) = 6 e ^ { - x / 3 }$ for all $x$. (a) Find the first four nonzero terms and the general term for the Taylor series for $f$ about $x = 0$. (b) Let $g$ be the function given by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$. Find the first four nonzero terms and the general term for the Taylor series for $g$ about $x = 0$. (c) The function $h$ satisfies $h ( x ) = k f ^ { \prime } ( a x )$ for all $x$, where $a$ and $k$ are constants. The Taylor series for $h$ about $x = 0$ is given by $$h ( x ) = 1 + x + \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 3 } } { 3 ! } + \cdots + \frac { x ^ { n } } { n ! } + \cdots .$$ Find the values of $a$ and $k$.

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 $f$ is defined by the power series
$$f ( x ) = 1 + ( x + 1 ) + ( x + 1 ) ^ { 2 } + \cdots + ( x + 1 ) ^ { n } + \cdots = \sum _ { n = 0 } ^ { \infty } ( x + 1 ) ^ { n }$$
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 power series above is the Taylor series for $f$ about $x = - 1$. Find the sum of the series for $f$.
(c) Let $g$ be the function defined by $g ( x ) = \int _ { - 1 } ^ { x } f ( t ) d t$. Find the value of $g \left( - \frac { 1 } { 2 } \right)$, if it exists, or explain why $g \left( - \frac { 1 } { 2 } \right)$ cannot be determined.
(d) Let $h$ be the function defined by $h ( x ) = f \left( x ^ { 2 } - 1 \right)$. Find the first three nonzero terms and the general term of the Taylor series for $h$ about $x = 0$, and find the value of $h \left( \frac { 1 } { 2 } \right)$.

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 $\frac{1}{6!}$.

6. The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 2 x ) ^ { n } } { n - 1 }$ on its interval of convergence.
(a) Find the interval of convergence for the Maclaurin series of $f$. Justify your answer.
(b) Show that $y = f ( x )$ is a solution to the differential equation $x y ^ { \prime } - y = \frac { 4 x ^ { 2 } } { 1 + 2 x }$ for $| x | < R$, where $R$ is the radius of convergence from part (a).

WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM

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 $\left|P_4\left(\frac{1}{4}\right) - f\left(\frac{1}{4}\right)\right| < \frac{1}{3000}$.

Let $f ( x ) = \ln \left( 1 + x ^ { 3 } \right)$.
(a) The Maclaurin series for $\ln ( 1 + x )$ is $x - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } + \cdots + ( - 1 ) ^ { n + 1 } \cdot \frac { x ^ { n } } { n } + \cdots$. Use the series to write the first four nonzero terms and the general term of the Maclaurin series for $f$.
(b) The radius of convergence of the Maclaurin series for $f$ is 1 . Determine the interval of convergence. Show the work that leads to your answer.
(c) Write the first four nonzero terms of the Maclaurin series for $f ^ { \prime } \left( t ^ { 2 } \right)$. If $g ( x ) = \int _ { 0 } ^ { x } f ^ { \prime } \left( t ^ { 2 } \right) d t$, use the first two nonzero terms of the Maclaurin series for $g$ to approximate $g ( 1 )$.
(d) The Maclaurin series for $g$, evaluated at $x = 1$, is a convergent alternating series with individual terms that decrease in absolute value to 0 . Show that your approximation in part (c) must differ from $g ( 1 )$ by less than $\frac { 1 } { 5 }$.

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