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3 years ago

I need help is it A B C or D

Mathematics

1 answer:

777dan777 [17]3 years ago

4 0

The answer is D, thanks dude who corrected me!

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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

Solve 15n > 12n + 18

gogolik [260]

15n > 12n + 18
3n > 18
n > 6

3 0

3 years ago

Read 2 more answers

5-2×=13 cual es el numero para 2x

olya-2409 [2.1K]

Answer: 39

Step-by-step explanation: x=13. so 5-2=3 x 13 = 39 espero que esto ayude :)

3 0

3 years ago

Write or draw to explain a way to find each sum: 6+7, 8+4, 2+9

storchak [24]

Draw six dots then a plus sign then draw seven dots and the count them and put an equal sign and however many dots. Doe the same with the other numbers

6 0

2 years ago

Read 2 more answers

the vertex of this parabola is at (3,5). When the y-value is 6, the x-value is -1. What is the coefficient of the squared term i

otez555 [7]

Now, assuming is a vertical parabola, so the squared variable is the "x", let's do some checking.

$\bf \qquad \textit{parabola vertex form}\\\\
\begin{array}{llll}
\boxed{y=a(x-{{ h}})^2+{{ k}}}\\\\
x=a(y-{{ k}})^2+{{ h}}
\end{array} \qquad\qquad vertex\ ({{ h}},{{ k}})\\\\
-------------------------------\\\\
vertex
\begin{cases}
h=3\\
k=5
\end{cases}\implies y=a(x-3)^2+5
\\\\\\
\textit{we also know that }
\begin{cases}
y=6\\
x=-1
\end{cases}\implies 6=a(-1-3)^2+5
\\\\\\
1=a(-4)^2\implies 1=16a\implies \cfrac{1}{16}=a$

6 0

3 years ago