I need help solving this please?

Use the definition of a Taylor series to find the first three non zero terms of the Taylor series for the given function centere

Ket [755]

Answer:

$e^{4x}=e^4+4e^4(x-1)+8e^4(x-1)^2+...$

$\displaystyle e^{4x}=\sum^{\infty}_{n=0} \dfrac{4^ne^4}{n!}(x-1)^n$

Step-by-step explanation:

Taylor series expansions of f(x) at the point x = a

$\text{f}(x)=\text{f}(a)+\text{f}\:'(a)(x-a)+\dfrac{\text{f}\:''(a)}{2!}(x-a)^2+\dfrac{\text{f}\:'''(a)}{3!}(x-a)^3+...+\dfrac{\text{f}\:^{(r)}(a)}{r!}(x-a)^r+...$

This expansion is valid only if $\text{f}\:^{(n)}(a)$ exists and is finite for all $n \in \mathbb{N}$ , and for values of x for which the infinite series converges.

$\textsf{Let }\text{f}(x)=e^{4x} \textsf{ and }a=1$

$\text{f}(x)=\text{f}(1)+\text{f}\:'(1)(x-1)+\dfrac{\text{f}\:''(1)}{2!}(x-1)^2+...$

$\boxed{\begin{minipage}{5.5 cm}\underline{Differentiating $e^{f(x)}$}\\\\If $y=e^{f(x)}$, then $\dfrac{\text{d}y}{\text{d}x}=f\:'(x)e^{f(x)}$\\\end{minipage}}$

$\text{f}(x)=e^{4x} \implies \text{f}(1)=e^4$

$\text{f}\:'(x)=4e^{4x} \implies \text{f}\:'(1)=4e^4$

$\text{f}\:''(x)=16e^{4x} \implies \text{f}\:''(1)=16e^4$

Substituting the values in the series expansion gives:

$e^{4x}=e^4+4e^4(x-1)+\dfrac{16e^4}{2}(x-1)^2+...$

Factoring out e⁴:

$e^{4x}=e^4\left[1+4(x-1)+8}(x-1)^2+...\right]$

Taylor Series summation notation:

$\displaystyle \text{f}(x)=\sum^{\infty}_{n=0} \dfrac{\text{f}\:^{(n)}(a)}{n!}(x-a)^n$

Therefore:

$\displaystyle e^{4x}=\sum^{\infty}_{n=0} \dfrac{4^ne^4}{n!}(x-1)^n$

7 0

1 year ago

The O! Natural Company sells a juice in 1 gallon bottles. The current retail price of the juice is $3.50 for 1 gallon. In order

cluponka [151]

Answer:

After the discount, the customer will pay only 91.4% of the initial price.

Step-by-step explanation:

We have that the price of the current retailer, in dollars, is:

3.50

After the discount, the new price, in dollars, is:

3.20

We want to know what percentage of the original price is the final price.

To find out, we must divide the final price between the initial price and then multiply the result by 100%

So:

$\frac{
3.20}{3.50}$ *100% = 0.914. * 100% = 91.4%

After the discount, the customer will pay only 91.4% of the initial price.

The discount percentage you are going to pay is 100% -91.4% = 8.6% of the initial price

8 0

3 years ago

What is 1.06 as a mixed number in simplest form?

Maru [420]

Okay so you have the question What is 1.06 as a mixed number in simplest form?
So, let us begin okay so 1.06 = 106/100 which then we will simplify to 53/50 and then you can simplify it one more time all the way down to it's simplest form of 1 3/50

Hope i helped
Cheers,
Belive1234

3 0

2 years ago

Read 2 more answers

12=15(z−51) What is Z

professor190 [17]

Answer:

51.8

Step-by-step explanation:

Multiply 15 by z and -51 which would equal (15z-765).

Add 765 to 12 and that equals 777

Divide 777 by 15 which would equal 51.8

4 0