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.

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 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 $f(1.4)$.

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 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.

Let $n \in \mathbb{N}$. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Determine $L_0, L_1, L_2$ and $L_3$.