What is this question asking and how do you solve it?

What is the question asking?

When a function is differentiable, and we know the value of $y$ at a fixed point $x_0$, we can approximate the approximate change in $y$ as $x$ changes a little bit. The formula is:

change in $y \approx \dfrac{dy}{dx}(x_0) \times$ change in $x$

What is the actual answer to the question:

The derivative of $y$ with respect to $x$ is:

$\dfrac{dy}{dx} = \dfrac{1}{2}e^{x/2} + \cos (2x) - 2x\sin (2x)$

The approximate change in $y$ when $x$ changes from $1$ to $1+h$ is:

$y(1+h)-y(1) = \dfrac{dy}{dx}(1)\times (1+h-1) =\left( \dfrac{1}{2}e^{1/2} + \cos (2) - 2\sin (2) \right) \times h$