Power Series

What is a Taylor polynomial for a function f at a point $x_0$?

Let f be a twice-differentiable function at a point $x_0$. Using the words value, slope, and concavity, explain why the second Taylor polynomial $P_2\left(x\right)$ might be a good approximation for f close to $x_0$.

Let f be a twice-differentiable function at a point $x_0$. Explain why the sum

$f\left(x\right) + f\text{'}\left(x\right) (x-x_0)+\frac{f\text{'}\text{'}\left(x\right)}{2!}{(x-x_0)}^2$

is not the second-order Taylor polynomial for f at $x_0$.

What is a difference between the Maclaurin polynomial of order n and the Taylor polynomial of order n for a function f ?

What is a difference between a Maclaurin polynomial and the Maclaurin series for a function f ?

What is a difference between a Taylor polynomial and the Taylor series for a function f at a point $x_0$?

What is the relationship between a Maclaurin series and a power series in x?

If a function f has a Maclaurin series, what are the possibilities for the interval of convergence for that series?

If a function f has a Taylor series at $x_0$, what are the possibilities for the interval of convergence for that series?

Let $f\left(x\right)=3x^2-2x+5$. Find the first-, second-, and third-order Maclaurin polynomials, $P_1\left(x\right)$, $P_2\left(x\right)$, and $P_3\left(x\right)$, for $f$ . Explain why $f\left(x\right)=P_2\left(x\right)=P_3\left(x\right)$. Graph $f\left(x\right)$, $P_1\left(x\right)$, and $P_2\left(x\right)$.

Let $f\left(x\right)=4x^3-5x^2-6x+7$. Find the first- through fourth-order Maclaurin polynomials, $P_1\left(x\right), P_2\left(x\right), P_3\left(x\right),$ and $P_4\left(x\right)$, for $f$ . Explain why $f\left(x\right)=P_3\left(x\right)=P_4\left(x\right)$. Graph $f\left(x\right), P_1\left(x\right), P_2\left(x\right)$, and $P_3\left(x\right)$.

Let $f\left(x\right)=3x^2-2x+5$. Find the first-, second-. and third-order Taylor polynomials, $P_1\left(x\right), P_2\left(x\right),$ and $P_3\left(x\right)$, for $f$ at $1$. Explain why $f\left(x\right)=P_2\left(x\right)=P_3\left(x\right)$.

Let $f\left(x\right)=4x^3-5x^2-6x+7$. Find the first- through fourth-order Taylor polynomials, $P_1\left(x\right), P_2\left(x\right), P_3\left(x\right),$and $P_4\left(x\right)$, for $f$ at $1$. Explain why $f\left(x\right)=P_3\left(x\right)=P_4\left(x\right)$.

Let $f\left(x\right)=a{x}^3+b{x}^2+cx+d$, where $a,b,c,$ and $d$ are constants. Find the first- through fourth-order Taylor polynomials, $P_1\left(x\right), P_2\left(x\right), P_3\left(x\right),$ and $P_4\left(x\right)$, for $f$ at $x_0$. Explain why $f\left(x\right)=P_3\left(x\right)=P_4\left(x\right)$.

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$\cos \left(x\right)$.

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$e^x$.

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$\sin \left(x\right)$.

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$\ln \left(1+x\right)$.

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$\cos \left(2x\right)$.

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$\sin \left(3x\right)$.

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$\sqrt{1-x}$

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$\frac{1+x}{1-x}$.

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$x\sin \left(x\right)$.

Find the fourth Maclaurin polynomial $P_4\left(x\right)$ for the specified function:

$x^2{e}^x$.

Find the Maclaurin series for the specified function:

$\cos \left(x\right)$.

Find the Maclaurin series for the specified function:

$e^x$.

Find the Maclaurin series for the specified function:

$\sin \left(x\right)$.

In Exercises 31–40 find the Maclaurin series for the specified function. Note: These are the same functions as in Exercises 21–30.

$f\left(x\right)=\cos 2x$

In Exercises 31–40 find the Maclaurin series for the specified function. Note: These are the same functions as in Exercises 21–30.

$\sin 3x$

In Exercises 31–40 find the Maclaurin series for the specified function. Note: These are the same functions as in Exercises 21–30.

$\sqrt{1-x}$

In Exercises 31–40 find the Maclaurin series for the specified function. Note: These are the same functions as in Exercises 21–30.

$\frac{1+x}{1-x}$

In Exercises 31–40 find the Maclaurin series for the specified function. Note: These are the same functions as in Exercises 21–30.

$x \sin x$

In Exercises 31–40 find the Maclaurin series for the specified function. Note: These are the same functions as in Exercises 21–30.

$x^2{e}^x$

$P_4\left(x\right)$In Exercises 41–48 find the fourth Taylor polynomial  for the specified function and the given value of $x_0$.

$\cos x ,\frac{\mathrm{\pi }}{2}$

In Exercises 41–48 find the fourth Taylor polynomial $P_4\left(x\right)$ for the specified function and the given value of $x_0$.

$e^{x },1$

In Exercises 41–48 find the fourth Taylor polynomial $P_4\left(x\right)$ for the specified function and the given value of $x_0$.

$Sin x,\mathrm{\pi }$

In Exercises 41–48 find the fourth Taylor polynomial $P_4\left(x\right)$ for the specified function and the given value of $x_0$.

$\sqrt{x} ,1$

In Exercises 41–48 find the fourth Taylor polynomial $P_4\left(x\right)$ for the specified function and the given value of $x_0$.

$45. \ln \left(x\right),3$

In Exercises 41–48 find the fourth Taylor polynomial $P_4\left(x\right)$ for the specified function and the given value of $x_0$

$46. \sqrt[3]{x},1$

In Exercises 41–48 find the fourth Taylor polynomial $P_4\left(x\right)$ for the specified function and the given value of $x_0$.

$\text{47. }\cos 2x,\frac{\pi }{4}$

In Exercises 41–48 find the fourth Taylor polynomial $P_4\left(x\right)$ for the specified function and the given value of $x_0$.

$\text{48. }\sin 3x,\frac{\pi }{6}$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$. Note: These are the same functions and values as in Exercises 41–48.

$49. \cos \left(x\right),\frac{\pi }{2}$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$. Note: These are the same functions and values as in Exercises 41–48.

$50. e^x,1$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$. Note: These are the same functions and values as in Exercises 41–48.

$51. \sin \left(x\right),\mathrm{\pi }$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$. Note: These are the same functions and values as in Exercises 41–48.

$52. \sqrt{x},1$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$. Note: These are the same functions and values as in Exercises 41–48.

$53. \ln \left(x\right),3$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$. Note: These are the same functions and values as in Exercises 41–48.

$54. \sqrt[3]{x},1$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$. Note: These are the same functions and values as in Exercises 41–48.

$55. \cos 2x,\frac{\pi }{4}$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$.

56. $\sin 3x,\frac{\mathrm{\pi }}{6}$

Show that if $p$is a positive integer, then the binomial series for $f\left(x\right)={(1+x)}^p$is a polynomial.