Question 1

Solve $1+x+x^{2}+x^{3}+\cdots=8$ for x.Round to 2 decimal places.

Accepted Answer

0.88

Question 2

Find the Maclaurin series for $f(x)=\sin (5 x)$ .

Accepted Answer

A)  $5 x-\frac{5^{3} x^{3}}{3 !}+\frac{5^{5} x^{5}}{5 !}-$ 
B)  $1-\frac{5^{2} x^{2}}{2 !}+\frac{5^{4} x^{4}}{4 !}-\cdots$ 
C)  $x-\frac{5^{2} x^{3}}{3 !}+\frac{5^{4} x^{5}}{5 !}-\cdots$ 
D)  $5 x-\frac{5^{2} x^{2}}{2 !}+\frac{5^{3} x^{3}}{3 !}-\cdots$ 
A)  $5 x-\frac{5^{3} x^{3}}{3 !}+\frac{5^{5} x^{5}}{5 !}-$ 
B)  $1-\frac{5^{2} x^{2}}{2 !}+\frac{5^{4} x^{4}}{4 !}-\cdots$ 
C)  $x-\frac{5^{2} x^{3}}{3 !}+\frac{5^{4} x^{5}}{5 !}-\cdots$ 
D)  $5 x-\frac{5^{2} x^{2}}{2 !}+\frac{5^{3} x^{3}}{3 !}-\cdots$ A

Question 3

The function h(x)is a continuous differentiable function whose graph is drawn below.The accompanying table provides some information about h(x)and its derivatives.   $$\begin{array} { c c c c c } 
\boldsymbol { x } & \boldsymbol { h } ( \boldsymbol { x } ) & \boldsymbol { h } ^ { \prime } ( \boldsymbol { x } ) & \boldsymbol { h } ^{ { \prime }{ \prime }}  ( \boldsymbol { x } ) & \boldsymbol { h } ^ {{ \prime }{ \prime } { \prime } }  ( \boldsymbol { x } ) \
0 & 2 & 1 & 0.50 & 0.25 \
1 & 3.29 & 1.64 & 0.82 & 0.41 \
2 & 5.43 & 2.71 & 1.35 & 0.67 \
3 & 8.96 & 4.48 & 2.24 & 1.12
\end{array}$$ h(x), h'(x), h"(x)and h"'(x)are all increasing functions.Suppose we use a tangent line approximation at zero to approximate h(0.1).Find a good upper bound for the error.

Accepted Answer

A) 0.0025 
B) 0.0820 
C) 0.1066 
D) 0.1558 
A) 0.0025 
B) 0.0820 
C) 0.1066 
D) 0.1558 B

Question 4

Use the derivative of the Taylor series about 0 for $\frac{1}{1-x}$ to find the Taylor series about 0 for $\frac{4 x}{(1-x)^{2}}$ .

Accepted Answer

A)  $4+4 x^{3}+4 x^{5}+4 x^{7}+\cdots$ 
B)  $4+4 x^{2}+4 x^{3}+4 x^{4}+\cdots$ 
C)  $4 x+8 x^{2}+12 x^{3}+16 x^{4}+\cdots$ 
D)  $4+8 x+12 x^{2}+16 x^{3}+\cdots$ 
A)  $4+4 x^{3}+4 x^{5}+4 x^{7}+\cdots$ 
B)  $4+4 x^{2}+4 x^{3}+4 x^{4}+\cdots$ 
C)  $4 x+8 x^{2}+12 x^{3}+16 x^{4}+\cdots$ 
D)  $4+8 x+12 x^{2}+16 x^{3}+\cdots$

Question 5

Find an expression for the general term of the Taylor series for $x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}$ .

Accepted Answer

A)  $\frac{(-1)^{n-1} x^{2 n-1}}{(2 n+1) !}$ 
B)  $(-1)^{2 n-1} \frac{x^{n+1}}{n !}$ 
C)  $(-1)^{n} \frac{x^{2 n+2}}{(2 n+1) !}$ 
D)  $\frac{x^{2 n}}{2 n-1}$ 
A)  $\frac{(-1)^{n-1} x^{2 n-1}}{(2 n+1) !}$ 
B)  $(-1)^{2 n-1} \frac{x^{n+1}}{n !}$ 
C)  $(-1)^{n} \frac{x^{2 n+2}}{(2 n+1) !}$ 
D)  $\frac{x^{2 n}}{2 n-1}$

Question 6

Find the Taylor series centered at $x=0$ (i.e.the Maclaurin series)for $\sin \left(x^{2}\right)$ .

Accepted Answer

A)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+3}}{(4 i+1) !}$ 
B)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+2}}{(2 i+1) !}$ 
C)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i}}{(2 i) !}$ 
D)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+1}}{(4 i) !}$ 
A)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+3}}{(4 i+1) !}$ 
B)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+2}}{(2 i+1) !}$ 
C)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i}}{(2 i) !}$ 
D)  $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{4 i+1}}{(4 i) !}$

Question 7

According to the theory of relativity, the energy, E, of a body of mass m is given as a function of its speed, v, by $E=m c^{2}\left(\frac{1}{\sqrt{1-v^{2} / c^{2}}}-1\right)$ , where c is a constant, the speed of light.Assuming v < c, expand E as a series in v/c, as far as the second non-zero term.

Accepted Answer

A)  $m c^{2}\left[1+\frac{1}{2} \frac{v^{2}}{c^{2}}+\cdots\right]$ 
B)  $m c^{2}\left[\frac{1}{2} \frac{v}{c}+\frac{3}{4} \frac{v^{2}}{c^{2}}+\cdots\right]$ 
C)  $m c^{2}\left[1+\frac{1}{2} \frac{v}{c}+\cdots\right]$ 
D)  $m c^{2}\left[\frac{1}{2} \frac{v^{2}}{c^{2}}+\frac{3}{8} \frac{v^{4}}{c^{4}}+\cdots\right]$ 
A)  $m c^{2}\left[1+\frac{1}{2} \frac{v^{2}}{c^{2}}+\cdots\right]$ 
B)  $m c^{2}\left[\frac{1}{2} \frac{v}{c}+\frac{3}{4} \frac{v^{2}}{c^{2}}+\cdots\right]$ 
C)  $m c^{2}\left[1+\frac{1}{2} \frac{v}{c}+\cdots\right]$ 
D)  $m c^{2}\left[\frac{1}{2} \frac{v^{2}}{c^{2}}+\frac{3}{8} \frac{v^{4}}{c^{4}}+\cdots\right]$

Question 8

Find the second harmonic of the function $h(x)=$$\left\{\begin{array}{c}
2 \pi \
0
\end{array}\right.$$\begin{aligned}
-\pi< \
0<
\end{aligned}$$\begin{array}{l}
x \leq 0 \
x \leq \pi
\end{array}$ .

Accepted Answer

A)  $\frac{4}{3} \sin 3 x$ 
B)  $-\frac{4}{3} \sin 3 x$ 
C)  $\frac{2}{3} \sin 3 x$ 
D)  $-\frac{2}{3} \sin 3 x$ 
E) 0 
A)  $\frac{4}{3} \sin 3 x$ 
B)  $-\frac{4}{3} \sin 3 x$ 
C)  $\frac{2}{3} \sin 3 x$ 
D)  $-\frac{2}{3} \sin 3 x$ 
E) 0

Question 9

Find $a_{0}$ for the function $f(x)=$$\left\{\begin{array}{c}
-1 \
1
\end{array}\right.$$\begin{array}{c}
-\pi< \
0<
\end{array}$$\begin{array}{l}
x \leq 0 \
x \leq \pi
\end{array}$
 .

Accepted Answer

A) 1 
B) -1 
C) 1/2 
D) -1/2 
E) 0 
A) 1 
B) -1 
C) 1/2 
D) -1/2 
E) 0

Question 10

The function g has the Taylor approximation $g(x)=c_{0}+c_{1}(x-a)+c_{2}(x-a)^{2}$ and the graph given below:   Is c0 positive, negative, or zero?

Accepted Answer

Question 11

Use the first three nonzero terms of the Taylor polynomial to approximate $\int_{0.1}^{1.1} \frac{\sin x}{x} d x$ .Give your answer to 5 decimal places.

Accepted Answer

A) 0.92868 
B) 0.92880 
C) -0.92880 
D) -0.92868 
E) does not exist 
A) 0.92868 
B) 0.92880 
C) -0.92880 
D) -0.92868 
E) does not exist

Question 12

Find the Taylor polynomial of degree 3 around x = 0 for the function $f(x)=\sqrt{1-x}$ and use it to approximate $\sqrt{0.7}$ .Give your answer to 4 decimal places.

Accepted Answer

The answer of Find the Taylor polynomial of degree 3...

Question 13

Find the fourth term of the Taylor series for the function $f(x)=\cos x$ about $x=\pi / 3$ .

Accepted Answer

A)  $\frac{\left(x-\frac{\pi}{3}\right)^{3}}{12}$ 
B)  $-\frac{\left(x-\frac{\pi}{3}\right)^{3}}{12}$ 
C)  $\frac{\sqrt{3}\left(x-\frac{\pi}{3}\right)^{3}}{12}$ 
D)  $-\frac{\sqrt{3}\left(x-\frac{\pi}{3}\right)^{3}}{12}$ 
A)  $\frac{\left(x-\frac{\pi}{3}\right)^{3}}{12}$ 
B)  $-\frac{\left(x-\frac{\pi}{3}\right)^{3}}{12}$ 
C)  $\frac{\sqrt{3}\left(x-\frac{\pi}{3}\right)^{3}}{12}$ 
D)  $-\frac{\sqrt{3}\left(x-\frac{\pi}{3}\right)^{3}}{12}$

Question 14

Construct the Taylor polynomial approximation of degree 3 to the function $f(x)=\arctan x$ about the point x = 0.Use it to approximate the value $f(0.35)$ to 5 decimal places.How does the approximation compare to the actual value?

Accepted Answer

0.33571 The calculat

Question 15

Fill in the blanks: Fourier polynomials give good __________ approximations to a function.Taylor polynomials give good _____________ approximations to a function.

Accepted Answer

The answer of Fill in the blanks: Fourier polynomials give...

Question 16

Use the Maclaurin series for $f(x)=\sin (4 x)$ to find the Maclaurin series for $f(x)=\cos (4 x)$ .

Accepted Answer

A)  $4 x-\frac{4^{3} x^{3}}{3 !}+\frac{4^{5} x^{5}}{5 !}-\cdots$ 
B)  $1-\frac{4^{2} x^{2}}{2 !}+\frac{4^{4} x^{4}}{4 !}-\cdots$ 
C)  $x-\frac{4^{2} x^{3}}{3 !}+\frac{4^{4} x^{5}}{5 !}-\cdots$ 
D)  $4 x-\frac{4^{2} x^{2}}{2 !}+\frac{4^{3} x^{3}}{3 !}-\cdots$ 
A)  $4 x-\frac{4^{3} x^{3}}{3 !}+\frac{4^{5} x^{5}}{5 !}-\cdots$ 
B)  $1-\frac{4^{2} x^{2}}{2 !}+\frac{4^{4} x^{4}}{4 !}-\cdots$ 
C)  $x-\frac{4^{2} x^{3}}{3 !}+\frac{4^{4} x^{5}}{5 !}-\cdots$ 
D)  $4 x-\frac{4^{2} x^{2}}{2 !}+\frac{4^{3} x^{3}}{3 !}-\cdots$

Question 17

Use the binomial series to find the coefficient of the $x^{2}$ term in the expansion of $(1+x)^{4}$ .

Accepted Answer

The answer of Use the binomial series to find the...

Question 18

The graph of the function $f(x)=e^{-x^{2} / 5}$ is a bell-shaped curve similar to a normal probability density function.Is $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i}}{5^{i}(i !)}$ the Maclaurin series for $f(x)$ ?

Accepted Answer

A) True 
 B)False

Question 19

Recognize $5-\frac{5^{3}}{3 !}+\frac{5^{5}}{5 !}-\frac{5^{7}}{7 !}+\cdots$ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

Accepted Answer

The answer of Recognize $5-\frac{5^{3}}{3 !}+\frac{5^{5}}{5 !}-\frac{5^{7}}{7 !}+\cdots$ as a...

Question 20

Find the Taylor series for $\sin x-\cos x$ .

Accepted Answer

A)  $1+x-\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\frac{x^{5}}{5 !}-\cdots$ 
B)  $1-x-\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}-\frac{x^{5}}{5 !}-\cdots$ 
C)  $-1+x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}-\frac{x^{4}}{4 !}+\frac{x^{5}}{5 !}+\ldots$ 
D)  $-1-x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}-\frac{x^{4}}{4 !}-\frac{x^{5}}{5 !}+\ldots$ 
A)  $1+x-\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\frac{x^{5}}{5 !}-\cdots$ 
B)  $1-x-\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}-\frac{x^{5}}{5 !}-\cdots$ 
C)  $-1+x+\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}-\frac{x^{4}}{4 !}+\frac{x^{5}}{5 !}+\ldots$ 
D)  $-1-x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}-\frac{x^{4}}{4 !}-\frac{x^{5}}{5 !}+\ldots$

Solve $1+x+x^{2}+x^{3}+\cdots=8$ for x.Round to 2 decimal places.

Find the Maclaurin series for $f(x)=\sin (5 x)$ .

Use the derivative of the Taylor series about 0 for $\frac{1}{1-x}$ to find the Taylor series about 0 for $\frac{4 x}{(1-x)^{2}}$ .

Find an expression for the general term of the Taylor series for $x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}$ .

Find the Taylor series centered at $x=0$ (i.e.the Maclaurin series)for $\sin \left(x^{2}\right)$ .

According to the theory of relativity, the energy, E, of a body of mass m is given as a function of its speed, v, by $E=m c^{2}\left(\frac{1}{\sqrt{1-v^{2} / c^{2}}}-1\right)$ , where c is a constant, the speed of light.Assuming v < c, expand E as a series in v/c, as far as the second non-zero term.

Find the second harmonic of the function $h(x)=$ $\left\{\begin{array}{c}2 \pi \\0\end{array}\right.$ -\pi< 0< x\leq0 x\leq\pi .

Find $a_{0}$ for the function $f(x)=$ $\left\{\begin{array}{c}-1 \\1\end{array}\right.$ -\pi< 0< x\leq0 x\leq\pi .

Use the first three nonzero terms of the Taylor polynomial to approximate $\int_{0.1}^{1.1} \frac{\sin x}{x} d x$ .Give your answer to 5 decimal places.

Find the Taylor polynomial of degree 3 around x = 0 for the function $f(x)=\sqrt{1-x}$ and use it to approximate $\sqrt{0.7}$ .Give your answer to 4 decimal places.

Find the fourth term of the Taylor series for the function $f(x)=\cos x$ about $x=\pi / 3$ .

Construct the Taylor polynomial approximation of degree 3 to the function $f(x)=\arctan x$ about the point x = 0.Use it to approximate the value $f(0.35)$ to 5 decimal places.How does the approximation compare to the actual value?

Fill in the blanks: Fourier polynomials give good approximations to a function.Taylor polynomials give good ___ approximations to a function.

Use the Maclaurin series for $f(x)=\sin (4 x)$ to find the Maclaurin series for $f(x)=\cos (4 x)$ .

Use the binomial series to find the coefficient of the $x^{2}$ term in the expansion of $(1+x)^{4}$ .

The graph of the function $f(x)=e^{-x^{2} / 5}$ is a bell-shaped curve similar to a normal probability density function.Is $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i}}{5^{i}(i !)}$ the Maclaurin series for $f(x)$ ?

Recognize $5-\frac{5^{3}}{3 !}+\frac{5^{5}}{5 !}-\frac{5^{7}}{7 !}+\cdots$ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

Find the Taylor series for $\sin x-\cos x$ .

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 10: Approximating Functions Using Series

Solve 1+x+x2+x3+⋯=81+x+x^{2}+x^{3}+\cdots=81+x+x2+x3+⋯=8 for x.Round to 2 decimal places.

Find the Maclaurin series for f(x)=sin⁡(5x)f(x)=\sin (5 x)f(x)=sin(5x) .

Use the derivative of the Taylor series about 0 for 11−x\frac{1}{1-x}1−x1​ to find the Taylor series about 0 for 4x(1−x)2\frac{4 x}{(1-x)^{2}}(1−x)24x​ .

Find an expression for the general term of the Taylor series for xsin⁡x=x2−x43!+x65!−x87!x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}xsinx=x2−3!x4​+5!x6​−7!x8​ .

Find the Taylor series centered at x=0x=0x=0 (i.e.the Maclaurin series)for sin⁡(x2)\sin \left(x^{2}\right)sin(x2) .

Find the second harmonic of the function h(x)=h(x)=h(x)={2π0\left\{\begin{array}{c}2 \pi \\0\end{array}\right.{2π0​ -\pi< 0< x\leq0 x\leq\pi .

Find a0a_{0}a0​ for the function f(x)=f(x)=f(x)={−11\left\{\begin{array}{c}-1 \\1\end{array}\right.{−11​ -\pi< 0< x\leq0 x\leq\pi .

The function g has the Taylor approximation g(x)=c0+c1(x−a)+c2(x−a)2g(x)=c_{0}+c_{1}(x-a)+c_{2}(x-a)^{2}g(x)=c0​+c1​(x−a)+c2​(x−a)2 and the graph given below: Is c0 positive, negative, or zero?

Use the first three nonzero terms of the Taylor polynomial to approximate ∫0.11.1sin⁡xxdx\int_{0.1}^{1.1} \frac{\sin x}{x} d x∫0.11.1​xsinx​dx .Give your answer to 5 decimal places.

Find the Taylor polynomial of degree 3 around x = 0 for the function f(x)=1−xf(x)=\sqrt{1-x}f(x)=1−x​ and use it to approximate 0.7\sqrt{0.7}0.7​ .Give your answer to 4 decimal places.

Find the fourth term of the Taylor series for the function f(x)=cos⁡xf(x)=\cos xf(x)=cosx about x=π/3x=\pi / 3x=π/3 .

Construct the Taylor polynomial approximation of degree 3 to the function f(x)=arctan⁡xf(x)=\arctan xf(x)=arctanx about the point x = 0.Use it to approximate the value f(0.35)f(0.35)f(0.35) to 5 decimal places.How does the approximation compare to the actual value?

Fill in the blanks: Fourier polynomials give good __________ approximations to a function.Taylor polynomials give good _____________ approximations to a function.

Use the Maclaurin series for f(x)=sin⁡(4x)f(x)=\sin (4 x)f(x)=sin(4x) to find the Maclaurin series for f(x)=cos⁡(4x)f(x)=\cos (4 x)f(x)=cos(4x) .

Use the binomial series to find the coefficient of the x2x^{2}x2 term in the expansion of (1+x)4(1+x)^{4}(1+x)4 .

The graph of the function f(x)=e−x2/5f(x)=e^{-x^{2} / 5}f(x)=e−x2/5 is a bell-shaped curve similar to a normal probability density function.Is ∑i=0∞(−1)ix2i5i(i!)\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i}}{5^{i}(i !)}∑i=0∞​5i(i!)(−1)ix2i​ the Maclaurin series for f(x)f(x)f(x) ?

Recognize 5−533!+555!−577!+⋯5-\frac{5^{3}}{3 !}+\frac{5^{5}}{5 !}-\frac{5^{7}}{7 !}+\cdots5−3!53​+5!55​−7!57​+⋯ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

Find the Taylor series for sin⁡x−cos⁡x\sin x-\cos xsinx−cosx .

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Sequences and Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Solve $1+x+x^{2}+x^{3}+\cdots=8$ for x.Round to 2 decimal places.

Find the Maclaurin series for $f(x)=\sin (5 x)$ .

Use the derivative of the Taylor series about 0 for $\frac{1}{1-x}$ to find the Taylor series about 0 for $\frac{4 x}{(1-x)^{2}}$ .

Find an expression for the general term of the Taylor series for $x \sin x=x^{2}-\frac{x^{4}}{3 !}+\frac{x^{6}}{5 !}-\frac{x^{8}}{7 !}$ .

Find the Taylor series centered at $x=0$ (i.e.the Maclaurin series)for $\sin \left(x^{2}\right)$ .

Find the second harmonic of the function $h(x)=$ $\left\{\begin{array}{c}2 \pi \\0\end{array}\right.$ -\pi< 0< x\leq0 x\leq\pi .

Find $a_{0}$ for the function $f(x)=$ $\left\{\begin{array}{c}-1 \\1\end{array}\right.$ -\pi< 0< x\leq0 x\leq\pi .

Use the first three nonzero terms of the Taylor polynomial to approximate $\int_{0.1}^{1.1} \frac{\sin x}{x} d x$ .Give your answer to 5 decimal places.

Find the Taylor polynomial of degree 3 around x = 0 for the function $f(x)=\sqrt{1-x}$ and use it to approximate $\sqrt{0.7}$ .Give your answer to 4 decimal places.

Find the fourth term of the Taylor series for the function $f(x)=\cos x$ about $x=\pi / 3$ .

Construct the Taylor polynomial approximation of degree 3 to the function $f(x)=\arctan x$ about the point x = 0.Use it to approximate the value $f(0.35)$ to 5 decimal places.How does the approximation compare to the actual value?

Fill in the blanks: Fourier polynomials give good approximations to a function.Taylor polynomials give good ___ approximations to a function.

Use the Maclaurin series for $f(x)=\sin (4 x)$ to find the Maclaurin series for $f(x)=\cos (4 x)$ .

Use the binomial series to find the coefficient of the $x^{2}$ term in the expansion of $(1+x)^{4}$ .

The graph of the function $f(x)=e^{-x^{2} / 5}$ is a bell-shaped curve similar to a normal probability density function.Is $\sum_{i=0}^{\infty} \frac{(-1)^{i} x^{2 i}}{5^{i}(i !)}$ the Maclaurin series for $f(x)$ ?

Recognize $5-\frac{5^{3}}{3 !}+\frac{5^{5}}{5 !}-\frac{5^{7}}{7 !}+\cdots$ as a Taylor series evaluated at a particular value of x and find the sum to 4 decimal places.

Find the Taylor series for $\sin x-\cos x$ .