Find taylor polynomial for $e^x\cos x$

Question

Let $f(x) = e^x\cos x$. Approximate the function $f$ through the taylor polynomial at point $x_0 = 0$ and find the degree of the polynomial so that the remainder in the interval $[-\frac{1}{100},\frac{1}{100}]$ is smaller than $10^{-8}$

Approach:

$f(x)= e^x\cos x$

$f'(x)= e^x(\cos x-\sin x)$

$f''(x) = -2e^x\sin x$

$f'''(x)=-2e^x(\sin x+\cos x)$

I don't know what to do next

What are you stuck on -- getting the rest of the Taylor polynomial, or finding the degree so that the remainder etc.? — BallBoy, Jan 10 '18 at 18:43
I can't a formula for the n-th polynomial. Plus, I can't figure out a way to Find the degree so that etc. — Basil Hallaq, Jan 10 '18 at 18:51
You need Taylor's theorem with remainder, and just compute enough terms so that the remainder is guaranteed to be small enough on the interval in question. — Angina Seng, Jan 10 '18 at 18:54
You may exploit that $\text{Re}$ and $\frac{d^n}{dx^n}$ commute. It is pretty clear what the MacLaurin series of $e^{(1+i)x}$ is. — Jack D'Aurizio, Jan 10 '18 at 19:21
https://math.stackexchange.com/questions/1752455/100-th-derivative-of-the-function-fx-ex-cosx/1752462#1752462 — lab bhattacharjee, May 02 '19 at 20:10

Mohammad Riazi-Kermani · Answer 1 · 2018-01-10T19:48:14.173

1

$$e^x=1+x+x^2/2+x^3/6+....$$ $$cosx=1-x^/2-x^4/24+......$$ Multiply to get $$e^x.cosx=1+x/2-x^3/12+.....$$

The formula for the error term involves the derivative of $ e^x.cosx$ and $n!$.

With these 3 terms the error condition is met. Thus the degree is 3

edited Jan 10 '18 at 19:48

answered Jan 10 '18 at 19:36

Mohammad Riazi-Kermani

68,728

score 0 · Answer 2 · answered Jan 10 '18 at 18:53

If you are to determine $n$-th derivative [ because the coefficient you are searching for the function $h : x \mapsto e^x\cos(x)$ ] is given for $n \in \mathbb{N}$ by $$ a_n=\frac{h^{(n)}\left(0\right)}{n!} $$

If $f$ et $g$ are differenciable enough, you can use for $n \in \mathbb{N}$ $$ \left(fg\right)^{\left(n\right)}=\sum_{k=0}^{n}\binom{n}{k}f^{\left(k\right)}g^{\left(n-k\right)} $$

What are cosine and exponential $n$-th derivative ?

Find taylor polynomial for $e^x\cos x$

2 Answers2