The twice-differentiable function $f$ is defined for all real numbers and satisfies the following conditions: $$f(0) = 2,\quad f'(0) = -4,\quad \text{and}\quad f''(0) = 3.$$ (a) The function $g$ is given by $g(x) = e^{ax} + f(x)$ for all real numbers, where $a$ is a constant. Find $g'(0)$ and $g''(0)$ in terms of $a$. Show the work that leads to your answers. (b) The function $h$ is given by $h(x) = \cos(kx)f(x)$ for all real numbers, where $k$ is a constant. Find $h'(x)$ and write an equation for the line tangent to the graph of $h$ at $x = 0$.
The twice-differentiable function $f$ is defined for all real numbers and satisfies the following conditions:
$$f(0) = 2,\quad f'(0) = -4,\quad \text{and}\quad f''(0) = 3.$$
(a) The function $g$ is given by $g(x) = e^{ax} + f(x)$ for all real numbers, where $a$ is a constant. Find $g'(0)$ and $g''(0)$ in terms of $a$. Show the work that leads to your answers.
(b) The function $h$ is given by $h(x) = \cos(kx)f(x)$ for all real numbers, where $k$ is a constant. Find $h'(x)$ and write an equation for the line tangent to the graph of $h$ at $x = 0$.