Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant. $)$

$$\frac{1}{1-x}, \quad n=3,5,7$$

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Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant. $)$

$$\frac{1}{1+x}, \quad n=4,6,8$$

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Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant. $)$

$$\sqrt{1+x}, \quad n=2,3,4$$

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Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant. $)$

$$\sqrt[3]{1-x}, \quad n=2,3,4$$

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Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant. $)$

$$\cos x, \quad n=2,4,6$$

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Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant. $)$

$$\ln (1+x), \quad n=5,7,9$$

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Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant. $)$

$$\arctan x, \quad n=3,4$$

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Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant. $)$

$$\tan x, \quad n=3,4$$

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Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant. $)$

$$\frac{1}{\sqrt{1+x}}, \quad n=2,3,4$$

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Find the Taylor polynomials of degree $n$ approximating the functions for $x$ near $0 .$ (Assume $p$ is a constant. $)$

$$(1+x)^{p}, \quad n=2,3,4$$

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Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.

$$\sqrt{1-x}, \quad a=0, \quad n=3$$

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Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.

$$e^{x}, \quad a=1, \quad n=4$$

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Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.

$$\frac{1}{1+x}, \quad a=2, \quad n=4$$

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Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.

$$\cos x, \quad a=\pi / 2, \quad n=4$$

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Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.

$$\sin x, \quad a=-\pi / 4, \quad n=3$$

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Find the Taylor polynomial of degree $n$ for $x$ near the given point $a$.

$$\ln \left(x^{2}\right), \quad a=1, \quad n=4$$

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The Taylor polynomial of degree 7 of $f(x)$ is given by

$$

P_{7}(x)=1-\frac{x}{3}+\frac{5 x^{2}}{7}+8 x^{3}-\frac{x^{5}}{11}+8 x^{7}

$$

Find the Taylor polynomial of degree 3 of $f(x)$

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The function $f(x)$ is approximated near $x=0$ by the third-degree Taylor polynomial

$$

P_{3}(x)=2-x-x^{2} / 3+2 x^{3}

$$

Give the value of

(a) $\quad f(0)$

(b) $f^{\prime}(0)$

(c) $\quad f^{\prime \prime}(0)$

(d) $\quad f^{\prime \prime \prime}(0)$

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Find the second-degree Taylor polynomial for $f(x)=$ $4 x^{2}-7 x+2$ about $x=0 .$ What do you notice?

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Find the third-degree Taylor polynomial for $f(x)=$ $$x^{3}+7 x^{2}-5 x+1$ about $x=0 .$$ What do you notice?

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(a) Based on your observations in Problems $19-20$ make a conjecture about Taylor approximations in the case when $f$ is itself a polynomial.

(b) Show that your conjecture is true.

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Find the value of $f^{(5)}(1)$ if $f(x)$ is approximated near $x=1$ by the Taylor polynomial

$$

p(x)=\sum_{n=0}^{10} \frac{(x-1)^{n}}{n !}

$$

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Find a simplified formula for $P_{5}(x),$ the fifth-degree Taylor polynomial approximating $f$ near $x=0$.

Use the values in the table.

$$\begin{array}{c|c|c|c|c|c}

\hline f(0) & f^{\prime}(0) & f^{\prime \prime}(0) & f^{\prime \prime \prime}(0) & f^{(4)}(0) & f^{(5)}(0) \\

\hline-3 & 5 & -2 & 0 & -1 & 4 \\

\hline

\end{array}$$

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Find a simplified formula for $P_{5}(x),$ the fifth-degree Taylor polynomial approximating $f$ near $x=0$.

Let $f(0)=-1$ and, for $n > 0, f^{(n)}(0)=-(-2)^{n}$

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Suppose $P_{2}(x)=a+b x+c x^{2}$ is the second-degree Taylor polynomial for the function $f$ about $x=0 .$ What can you say about the signs of $a, b,$ and $c$ if $f$ has the graph given below?

Suppose $P_{2}(x)=a+b x+c x^{2}$ is the second-degree Taylor polynomial for the function $f$ about $x=0 .$ What can you say about the signs of $a, b,$ and $c$ if $f$ has the graph given below?

(GRAPH CAN'T COPY)

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Suppose $P_{2}(x)=a+b x+c x^{2}$ is the second-degree Taylor polynomial for the function $f$ about $x=0 .$ What can you say about the signs of $a, b,$ and $c$ if $f$ has the graph given below?

(GRAPH CAN'T COPY)

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(GRAPH CAN'T COPY)

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(GRAPH CAN'T COPY)

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Use the Taylor approximation for $x$ near 0

$$

\sin x \approx x-\frac{x^{3}}{3 !}

$$

to explain why $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$

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Use the fourth-degree Taylor approximation for $x$ near 0

$$

\cos x \approx 1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}

$$

to explain why $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}$

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Use a fourth-degree Taylor approximation for $e^{h},$ for $h$ near $0,$ to evaluate the following limits. Would your answer be different if you used a Taylor polynomial of higher degree?

(a) $\lim _{h \rightarrow 0} \frac{e^{h}-1-h}{h^{2}}$

(b) $\lim _{h \rightarrow 0} \frac{e^{h}-1-h-\frac{h^{2}}{2}}{h^{3}}$

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If $f(2)=g(2)=h(2)=0,$ and $f^{\prime}(2)=h^{\prime}(2)=0$

$g^{\prime}(2)=22,$ and $f^{\prime \prime}(2)=3, g^{\prime \prime}(2)=5, h^{\prime \prime}(2)=7$

calculate the following limits. Explain your reasoning.

(a) $\lim _{x \rightarrow 2} \frac{f(x)}{h(x)}$

(b) $\lim _{x \rightarrow 2} \frac{f(x)}{g(x)}$

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One of the two sets of functions, $f_{1}, f_{2}, f_{3},$ or $g_{1}, g_{2}$ $g_{3},$ is graphed in Figure $10.8 ;$ the other set is graphed in Figure $10.9 .$ Points $A$ and $B$ each have $x=0 .$ Taylor polynomials of degree 2 approximating these functions near $x=0$ are as follows:

$$\begin{array}{ll}

f_{1}(x) \approx 2+x+2 x^{2} & g_{1}(x) \approx 1+x+2 x^{2} \\

f_{2}(x) \approx 2+x-x^{2} & g_{2}(x) \approx 1+x+x^{2} \\

f_{3}(x) \approx 2+x+x^{2} & g_{3}(x) \approx 1-x+x^{2}

\end{array}$$

(a) Which group of functions, the $f$ s or the $g$ s, is represented by each figure?

(b) What are the coordinates of the points $A$ and $B ?$

(c) Match each function with the graphs (I)-(III) in the appropriate figure.

(FIGURES CAN'T COPY)

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Derive the formulas given in the box on page 543 for the coefficients of the Taylor polynomial approximating a function $f$ for $x$ near $a$

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(a) Find and multiply the Taylor polynomials of degree 1 near $x=0$ for the two functions $f(x)=1 /(1-x)$ and $g(x)=1 /(1-2 x)$

(b) Find the Taylor polynomial of degree 2 near $x=0$ for the function $h(x)=f(x) g(x)$

(c) Is the product of the Taylor polynomials for $f(x)$ and $g(x)$ equal to the Taylor polynomial for the function $h(x) ?$

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(a) Find and multiply the Taylor polynomials of degree 1 near $x=0$ for the two functions $f(x)$ and $g(x)$

(b) Find the Taylor polynomial of degree 2 near $x=0$ for the function $h(x)=f(x) g(x)$

(c) Show that the product of the Taylor polynomials for $f(x)$ and $g(x)$ and the Taylor polynomial for the function $h(x)$ are the same if $f^{\prime \prime}(0) g(0)+$ $f(0) g^{\prime \prime}(0)=0$

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(a) Find the Taylor polynomial approximation of degree

4 about $x=0$ for the function $f(x)=e^{x^{2}}$

(b) Compare this result to the Taylor polynomial approximation of degree 2 for the function $f(x)=e^{x}$ about $x=0 .$ What do you notice?

(c) Use your observation in part (b) to write out the Taylor polynomial approximation of degree 20 for the function in part (a).

(d) What is the Taylor polynomial approximation of degree 5 for the function $f(x)=e^{-2 x} ?$

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The integral $\int_{0}^{1}(\sin t / t) d t$ is difficult to approximate using, for example, left Riemann sums or the trapezoid rule because the integrand $(\sin t) / t$ is not defined at $t=0 .$ However, this integral converges; its value is $0.94608 \ldots .$ Estimate the integral using Taylor polynomials for $\sin t$ about $t=0$ of

(a) Degree 3

(b) Degree 5

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Consider the equations $\sin x=0.2$ and $x-\frac{x^{3}}{3 !}=0.2$

(a) How many solutions does each equation have?

(b) Which of the solutions of the two equations are approximately equal? Explain.

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When we model the motion of a pendulum, we replace

the differential equation

$$

\frac{d^{2} \theta}{d t^{2}}=-\frac{g}{l} \sin \theta \quad \text { by } \quad \frac{d^{2} \theta}{d t^{2}}=-\frac{g}{l} \theta

$$

where $\theta$ is the angle between the pendulum and the vertical. Explain why, and under what circumstances, it is reasonable to make this replacement.

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(a) Using a graph, explain why the following equation has a solution at $x=0$ and another just to the right of $x=0$

$$

\cos x=1-0.1 x

$$

(b) Replace cos $x$ by its second-degree Taylor polynomial near 0 and solve the equation. Your answers are approximations to the solutions to the original equation at or near 0

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Explain what is wrong with the statement.

If $f(x)=\ln (2+x),$ then the second-degree Taylor polynomial approximating $f(x)$ near $x=0$ has a negative constant term.

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Explain what is wrong with the statement.

Let $f(x)=\frac{1}{1-x} .$ The coefficient of the $x$ term of the Taylor polynomial of degree 3 approximating $f(x)$ near $x=0$ is -1

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Give an example of:

A function $f(x)$ for which every Taylor polynomial approximation near $x=0$ involves only odd powers of $x$.

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A function $f(x)$ for which every Taylor polynomial approximation near $x=0$ involves only odd powers of $x$.

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Decide if the statements in Problems are true or false. Give an explanation for your answer.

If $f(x)$ and $g(x)$ have the same Taylor polynomial of degree 2 near $x=0,$ then $f(x)=g(x)$

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Decide if the statements in Problems are true or false. Give an explanation for your answer.

Using $\sin \theta \approx \theta-\theta^{3} / 3 !$ with $\theta=1^{\circ},$ we have $\sin \left(1^{\circ}\right) \approx$

$1-1^{3} / 6=5 / 6$

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Decide if the statements in Problems are true or false. Give an explanation for your answer.

The Taylor polynomial of degree 2 for $e^{x}$ near $x=5$ is $1+(x-5)+(x-5)^{2} / 2$.

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Decide if the statements in Problems are true or false. Give an explanation for your answer.

If the Taylor polynomial of degree 2 for $f(x)$ near $x=0$

is $P_{2}(x)=1+x-x^{2},$ then $f(x)$ is concave up near $x=0$

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Decide if the statements in Problems are true or false. Give an explanation for your answer.

The quadratic approximation to $f(x)$ for $x$ near 0 is better than the linear approximation for all values of $x$

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Decide if the statements in Problems are true or false. Give an explanation for your answer.

A Taylor polynomial for $f$ near $x=a$ touches the graph of $f$ only at $x=a$

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Decide if the statements in Problems are true or false. Give an explanation for your answer.

The linear approximation to $f(x)$ near $x=-1$ shows that if $f(-1)=g(-1)$ and $f^{\prime}(-1) < g^{\prime}(-1),$ then $f(x) < g(x)$ for all $x$ sufficiently close to -1 (but not equal to -1 ).

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Decide if the statements in Problems are true or false. Give an explanation for your answer.

The quadratic approximation to $f(x)$ near $x=-1$ shows that if $f(-1)=g(-1), f^{\prime}(-1)=g^{\prime}(-1),$ and $f^{\prime \prime}(-1) < g^{\prime \prime}(-1),$ then $f(x) < g(x)$ for all $x$ sufficiently close to -1 (but not equal to -1 ).

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