Question 1

Use power series operations to find the Taylor series at x = 0 for the given function.
-$$\frac { x ^ { 7 } } { 1 + 2 x }$$

Accepted Answer

The answer of Use power series operations to find the...

Question 2

Solve the problem.
-Use a graphical method to determine the approximate interval for which the second order Taylor polynomial for ln (1 + x) at x = 0 approximates ln (1 + x) with an absolute error of no more than 0.04.

Accepted Answer

The second order Taylor polynomial for ln (1 + x) at x = 0 is given by:

$$ P_2(x) = x - \frac{x^2}{2} $$

To find the interval where this polynomial approximates ln (1 + x) with an absolute error of no more than 0.04, we need to find the values of x where:

$$ \left|ln(1+x)-P_2(x)\right| \leq 0.04 $$

Substituting the expression for P2(x), we get:

$$ \left|\ln(1+x)-x+\frac{x^2}{2}\right| \leq 0.04 $$

Using a graphing calculator, we can graph the function y = ln(1+x) - x + (x^2)/2 and then use the trace function to find the x-values where the function is equal to ±0.04. Alternatively, we can use algebraic techniques such as the quadratic formula to solve for the roots of the inequality:

$$ \left(\ln(1+x)-x+\frac{x^2}{2}\right)^2 \leq 0.04^2 $$

This gives us:

$$ -0.4310 \leq x \leq 0.5525 $$

Therefore, the answer is D.

Question 3

Use power series operations to find the Taylor series at x = 0 for the given function.
-$$x ^ { 10 } \ln ( 1 + 8 x )$$

Accepted Answer

Explanation:Using the power series for $\ln(1+x)$ and the power series for $x^n$, we get$$x^{10}\ln(1+8x) = x^{10}\sum_{n=1}^\infty \frac{(-1)^{n-1}8^nx^n}{n} = \sum_{n=1}^\infty \frac{(-1)^{n-1}8^nx^{n+10}}{n}$$

Question 4

Use power series operations to find the Taylor series at x = 0 for the given function.
-$$\frac { 1 } { ( 1 - 5 x ) ^ { 2 } }$$

Accepted Answer

The answer of Use power series operations to find the...

Question 5

Use power series operations to find the Taylor series at x = 0 for the given function.
-$x ^ { 5 } e ^ { 4 x }$

Accepted Answer

The answer of Use power series operations to find the...

Question 6

Use substitution to find the Taylor series at x = 0 of the given function.
-$\ln \left( 1 + x ^ { 2 } \right)$

Accepted Answer

We know that 
$$
\ln(1+x) = \sum_{n=1}^\infty\frac{(-1)^{n-1}x^n}{n}
$$
Thus, substituting $x^2$ for $x$ gives
$$
\ln(1+x^2) = \sum_{n=1}^\infty\frac{(-1)^{n-1}(x^2)^n}{n}=\sum_{n=1}^\infty\frac{(-1)^{n-1}x^{2n}}{n}
$$
This matches the form of option D.

Question 7

Use power series operations to find the Taylor series at x = 0 for the given function.
-$$x ^ { 3 } \cos \pi x$$

Accepted Answer

The answer of Use power series operations to find the...

Question 8

Find the first four nonzero terms in the Maclaurin series for the function.
-$e ^ { - 6 x }$

Accepted Answer

The Maclaurin series for $e^{ax}$ is $1 + ax + \frac{(ax)^2}{2!} + \frac{(ax)^3}{3!} + \ldots$. Substituting $a = -6$, we get $1 - 6x + \frac{(-6x)^2}{2!} + \frac{(-6x)^3}{3!} + \ldots$, which simplifies to $1 - 6x + \frac{36x^2}{2!} + \frac{-216x^3}{3!} + \ldots$.

Question 9

Use power series operations to find the Taylor series at x = 0 for the given function.
-$\mathrm { x } ^ { 4 } \tan ^ { - 1 } ( 4 \mathrm { x } )$

Accepted Answer

The answer of Use power series operations to find the...

Question 10

Use power series operations to find the Taylor series at x = 0 for the given function.
-$$x ^ { 8 } e ^ { x }$$

Accepted Answer

The answer of Use power series operations to find the...

Question 11

Provide an appropriate response.
-Which of the following statements is false?
I. For a function $\mathrm { f } ( \mathrm { x } )$, the Taylor polynomial approximation can always be improved by increasing the degree of 1 polynomial.
II. Of all polynomials of degree less than or equal to $\mathrm { n }$, the Taylor polynomial of order $n$ gives the best approximi $f ( x )$.
III. The Taylor series at $x =$ a can be obtained by substituting $x - a$ for $x$ in the corresponding Maclaurin series.

Accepted Answer

The answer of Provide an appropriate response.
-Which of the following...

Question 12

Find the first four nonzero terms in the Maclaurin series for the function.
-$\ln \left( 1 + x ^ { 3 } \right)$

Accepted Answer

The Maclaurin series for ln(1+x) is given by:
ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ...
Thus, for ln(1+x^3), replacing x with x^3, we have:
ln(1+x^3) = x^3 - (x^3)^2/2 + (x^3)^3/3 - (x^3)^4/4 + ...
Simplifying and taking the first four nonzero terms, we get:
x^3 - x^6/2 + x^9/3 - x^12/4 + ...
Therefore, the correct choice is D.

Question 13

Use power series operations to find the Taylor series at x = 0 for the given function.
-$$\frac { x ^ { 7 } } { 1 - 3 x }$$

Accepted Answer

To find the Taylor series of the given function, we first need to express it as a power series. We can use partial fraction decomposition to write:

$$\frac{x^7}{1-3x} = x^7 \sum_{n=0}^\infty 3^n x^n = \sum_{n=0}^\infty 3^n x^{n+7}$$

This is the power series we need, so the coefficients are:

$$a_n = \begin{cases} 3^{n-7} & 	ext{if }n \geq 7 \ 0 & 	ext{if }n < 7 \end{cases}$$

Using the formula for the Taylor series coefficients, we have:

$$a_n = \frac{f^{(n)}(0)}{n!}$$

where $f(x) = \frac{x^7}{1-3x}$. Therefore, the Taylor series at $x=0$ is:

$$\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n = \sum_{n=0}^\infty \frac{a_n}{n!} x^n = \sum_{n=0}^\infty \frac{3^{n-7}}{n!} x^{n+7}$$

which is equivalent to D.

Question 14

Find the first four nonzero terms in the Maclaurin series for the function.
-$e ^ { 4 x } \sqrt { 1 + x }$

Accepted Answer

Since we need to find the first four nonzero terms in the Maclaurin series, we first write the Maclaurin series for each of the factors in the given function and then multiply them:
\begin{align*}
e^{4x} &= \sum_{n=0}^{\infty} \frac{(4x)^n}{n!} = 1+4x+\frac{(4x)^2}{2!}+\frac{(4x)^3}{3!}+\ldots \
\sqrt{1+x} &= \sum_{n=0}^{\infty} \binom{1/2}{n}x^n = 1+\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3+\ldots
\end{align*}
Multiplying the two series and collecting like terms, we get:
\begin{align*}
e^{4x}\sqrt{1+x} &= \left(1+4x+\frac{(4x)^2}{2!}+\frac{(4x)^3}{3!}+\ldotsight)\left(1+\frac{1}{2}x-\frac{1}{8}x^2+\frac{1}{16}x^3+\ldotsight) \
&= 1+\left(4+\frac{1}{2}ight)x+\left(\frac{16}{2!}+\frac{1}{2}\cdot4-\frac{1}{8}ight)x^2+\left(\frac{64}{3!}+\frac{16}{2!}\cdot\frac{1}{2}-\frac{1}{8}\cdot\frac{1}{2}ight)x^3+\ldots \
&=1+\frac{9}{2}x+\frac{79}{8}x^2+\frac{683}{48}x^3+\ldots
\end{align*}
Therefore, the first four nonzero terms in the Maclaurin series for the given function are $1+\frac{9}{2}x+\frac{79}{8}x^2+\frac{683}{48}x^3$. Hence, the answer is D.

Question 15

Find the first four nonzero terms in the Maclaurin series for the function.
-sin x cos x

Accepted Answer

The Maclaurin series for sin x is $x - \frac{x^3}{6} + \frac{x^5}{120} - \frac{x^7}{5040} + \ldots$ and for cos x is $1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \ldots$. Multiplying these series and collecting terms gives the series for sin x cos x as $x - \frac{2}{3}x^3 + \frac{2}{15}x^5 - \frac{4}{315}x^7 + \ldots$.

Question 16

Use substitution to find the Taylor series at x = 0 of the given function.
-$$e ^ { - 8 x }$$

Accepted Answer

The Taylor series of $e^x$ at $x = 0$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$. For $e^{-8x}$, substitute $-8x$ for $x$ in the series to get $\sum_{n=0}^{\infty} \frac{(-8x)^n}{n!}$, which simplifies to $\sum_{n=0}^{\infty} \frac{(-1)^n(8)^nx^n}{n!}$.

Question 17

Use power series operations to find the Taylor series at x = 0 for the given function.
-$x ^ { 7 } \sin x$

Accepted Answer

The answer of Use power series operations to find the...

Question 18

Use substitution to find the Taylor series at x = 0 of the given function.
-$$\ln \left( \frac { 1 + x ^ { 3 } } { 1 - x ^ { 3 } } \right)$$

Accepted Answer

The answer of Use substitution to find the Taylor series...

Question 19

Use power series operations to find the Taylor series at x = 0 for the given function.
-$\cos ^ { 2 } ( 2 x )$

Accepted Answer

The Taylor series for $\cos x$ is $\sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}$. Substituting $2x$ for $x$ gives the Taylor series for $\cos 2x$:$$\cos 2x = \sum_{n=0}^\infty \frac{(-1)^n (2x)^{2n}}{(2n)!} = \sum_{n=0}^\infty \frac{(-1)^n 4^n x^{2n}}{(2n)!}$$Squaring this series gives the Taylor series for $\cos^2 2x$:$$\cos^2 2x = \left(\sum_{n=0}^\infty \frac{(-1)^n 4^n x^{2n}}{(2n)!}\right)^2 = \frac{1}{2} + \sum_{n=0}^\infty \frac{(-1)^n 4^{2n} x^{2n}}{(2n)!}$$

Question 20

Solve the problem.
-You plan to estimate e by evaluating the Maclaurin series for $f ( x ) = e ^ { x }$ at $x = 1$. How many terms of the series would you have to add to be sure the estimate is good to 5 decimal places?

Accepted Answer

The answer of Solve the problem.
-You plan to estimate e...

Quiz 11: Infinite Sequences and Series

Use power series operations to find the Taylor series at x = 0 for the given function. - $\frac { x ^ { 7 } } { 1 + 2 x }$

Solve the problem. -Use a graphical method to determine the approximate interval for which the second order Taylor polynomial for ln (1 + x) at x = 0 approximates ln (1 + x) with an absolute error of no more than 0.04.

Use power series operations to find the Taylor series at x = 0 for the given function. - $x ^ { 10 } \ln ( 1 + 8 x )$

Use power series operations to find the Taylor series at x = 0 for the given function. - $\frac { 1 } { ( 1 - 5 x ) ^ { 2 } }$

Use power series operations to find the Taylor series at x = 0 for the given function. - $x ^ { 5 } e ^ { 4 x }$

Use substitution to find the Taylor series at x = 0 of the given function. - $\ln \left( 1 + x ^ { 2 } \right)$

Use power series operations to find the Taylor series at x = 0 for the given function. - $x ^ { 3 } \cos \pi x$

Find the first four nonzero terms in the Maclaurin series for the function. - $e ^ { - 6 x }$

Use power series operations to find the Taylor series at x = 0 for the given function. - $\mathrm { x } ^ { 4 } \tan ^ { - 1 } ( 4 \mathrm { x } )$

Use power series operations to find the Taylor series at x = 0 for the given function. - $x ^ { 8 } e ^ { x }$

Find the first four nonzero terms in the Maclaurin series for the function. - $\ln \left( 1 + x ^ { 3 } \right)$

Use power series operations to find the Taylor series at x = 0 for the given function. - $\frac { x ^ { 7 } } { 1 - 3 x }$

Find the first four nonzero terms in the Maclaurin series for the function. - $e ^ { 4 x } \sqrt { 1 + x }$

Find the first four nonzero terms in the Maclaurin series for the function. -sin x cos x

Use substitution to find the Taylor series at x = 0 of the given function. - $e ^ { - 8 x }$

Use power series operations to find the Taylor series at x = 0 for the given function. - $x ^ { 7 } \sin x$

Use substitution to find the Taylor series at x = 0 of the given function. - $\ln \left( \frac { 1 + x ^ { 3 } } { 1 - x ^ { 3 } } \right)$

Use power series operations to find the Taylor series at x = 0 for the given function. - $\cos ^ { 2 } ( 2 x )$

Solve the problem. -You plan to estimate e by evaluating the Maclaurin series for $f ( x ) = e ^ { x }$ at $x = 1$ . How many terms of the series would you have to add to be sure the estimate is good to 5 decimal places?

Quiz 11: Infinite Sequences and Series

Use power series operations to find the Taylor series at x = 0 for the given function. - x71+2x\frac { x ^ { 7 } } { 1 + 2 x }1+2xx7​

Solve the problem. -Use a graphical method to determine the approximate interval for which the second order Taylor polynomial for ln (1 + x) at x = 0 approximates ln (1 + x) with an absolute error of no more than 0.04.

Use power series operations to find the Taylor series at x = 0 for the given function. - x10ln⁡(1+8x) x ^ { 10 } \ln ( 1 + 8 x ) x10ln(1+8x)

Use power series operations to find the Taylor series at x = 0 for the given function. - 1(1−5x) 2\frac { 1 } { ( 1 - 5 x ) ^ { 2 } }(1−5x) 21​

Use power series operations to find the Taylor series at x = 0 for the given function. - x5e4xx ^ { 5 } e ^ { 4 x }x5e4x

Use substitution to find the Taylor series at x = 0 of the given function. - ln⁡(1+x2) \ln \left( 1 + x ^ { 2 } \right) ln(1+x2)

Use power series operations to find the Taylor series at x = 0 for the given function. - x3cos⁡πxx ^ { 3 } \cos \pi xx3cosπx

Find the first four nonzero terms in the Maclaurin series for the function. - e−6xe ^ { - 6 x }e−6x

Use power series operations to find the Taylor series at x = 0 for the given function. - x4tan⁡−1(4x) \mathrm { x } ^ { 4 } \tan ^ { - 1 } ( 4 \mathrm { x } ) x4tan−1(4x)

Use power series operations to find the Taylor series at x = 0 for the given function. - x8exx ^ { 8 } e ^ { x }x8ex

Find the first four nonzero terms in the Maclaurin series for the function. - ln⁡(1+x3) \ln \left( 1 + x ^ { 3 } \right) ln(1+x3)

Use power series operations to find the Taylor series at x = 0 for the given function. - x71−3x\frac { x ^ { 7 } } { 1 - 3 x }1−3xx7​

Find the first four nonzero terms in the Maclaurin series for the function. - e4x1+xe ^ { 4 x } \sqrt { 1 + x }e4x1+x​

Find the first four nonzero terms in the Maclaurin series for the function. -sin x cos x

Use substitution to find the Taylor series at x = 0 of the given function. - e−8xe ^ { - 8 x }e−8x

Use power series operations to find the Taylor series at x = 0 for the given function. - x7sin⁡xx ^ { 7 } \sin xx7sinx

Use substitution to find the Taylor series at x = 0 of the given function. - ln⁡(1+x31−x3) \ln \left( \frac { 1 + x ^ { 3 } } { 1 - x ^ { 3 } } \right) ln(1−x31+x3​)

Use power series operations to find the Taylor series at x = 0 for the given function. - cos⁡2(2x) \cos ^ { 2 } ( 2 x ) cos2(2x)

Solve the problem. -You plan to estimate e by evaluating the Maclaurin series for f(x) =exf ( x ) = e ^ { x }f(x) =ex at x=1x = 1x=1 . How many terms of the series would you have to add to be sure the estimate is good to 5 decimal places?

Use power series operations to find the Taylor series at x = 0 for the given function. - $\frac { x ^ { 7 } } { 1 + 2 x }$

Use power series operations to find the Taylor series at x = 0 for the given function. - $x ^ { 10 } \ln ( 1 + 8 x )$

Use power series operations to find the Taylor series at x = 0 for the given function. - $\frac { 1 } { ( 1 - 5 x ) ^ { 2 } }$

Use power series operations to find the Taylor series at x = 0 for the given function. - $x ^ { 5 } e ^ { 4 x }$

Use substitution to find the Taylor series at x = 0 of the given function. - $\ln \left( 1 + x ^ { 2 } \right)$

Use power series operations to find the Taylor series at x = 0 for the given function. - $x ^ { 3 } \cos \pi x$

Find the first four nonzero terms in the Maclaurin series for the function. - $e ^ { - 6 x }$

Use power series operations to find the Taylor series at x = 0 for the given function. - $\mathrm { x } ^ { 4 } \tan ^ { - 1 } ( 4 \mathrm { x } )$

Use power series operations to find the Taylor series at x = 0 for the given function. - $x ^ { 8 } e ^ { x }$

Find the first four nonzero terms in the Maclaurin series for the function. - $\ln \left( 1 + x ^ { 3 } \right)$

Use power series operations to find the Taylor series at x = 0 for the given function. - $\frac { x ^ { 7 } } { 1 - 3 x }$

Find the first four nonzero terms in the Maclaurin series for the function. - $e ^ { 4 x } \sqrt { 1 + x }$

Use substitution to find the Taylor series at x = 0 of the given function. - $e ^ { - 8 x }$

Use power series operations to find the Taylor series at x = 0 for the given function. - $x ^ { 7 } \sin x$

Use substitution to find the Taylor series at x = 0 of the given function. - $\ln \left( \frac { 1 + x ^ { 3 } } { 1 - x ^ { 3 } } \right)$

Use power series operations to find the Taylor series at x = 0 for the given function. - $\cos ^ { 2 } ( 2 x )$

Solve the problem. -You plan to estimate e by evaluating the Maclaurin series for $f ( x ) = e ^ { x }$ at $x = 1$ . How many terms of the series would you have to add to be sure the estimate is good to 5 decimal places?