Verify the given linear approximation at $a = 0$. Then determine the values of x for which the linear approximation is accurate to within 0.1.
So first I verified that the linear approximation is right.
$$e^x \cdot \cos x \approx 1 + x$$
Check
$$f'(x) = e^x \cdot (-\sin{x}) + \cos{x} \cdot e^x$$
$$\text{linear approx} = e^x \cdot \cos{x} + (e^x \cdot (-\sin{x}) + \cos{x} \cdot e^x)(x-a)$$
$$L(0) = 1 + 1(x-0) = 1+x$$
b. What are the values of x so that the linear approximation is within 0.1?
So I think I setup this equation correctly?
$$-0.1 < e^x\cos{x} – 1 – x < 0.1$$
Where do I go from here?
Best Answer
The next approximation would be the quadratic one, which would be of the form $1+x+\frac 12f''(0)x^2$. It will not be exact, but a very good approximation is to find the range of $x$ for which $\frac 12f''(0)x^2\lt 0.1$, so evaluate $f''(0)$ and plug in.