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
Best Answer
$$e^x=1+x+x^2/2+x^3/6+....$$ $$cosx=1-x^/2-x^4/24+......$$ 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