**Directions:**

- Show each step of your work and fully simplify each expression.
- Turn in your answers in class on a physical piece of paper.
**Staple multiple sheets together.**- Feel free to use Desmos for graphing.

Answer the following:

- Given the following function:
Find the sign of each of the following:
- $f'(x)$ on $(1, 3)$
- $f'(4)$
- $f'(6)$
- $f'(x)$ on $(6, 9)$
- $f'(12)$ (hint: derivative is a limit. when do limits exist?)
- $f'(x)$ on $(9, 12)$

- Again consider the function above. Determine the critical numbers and give the reason why they are a critical number.
- Find the intervals where each function is increasing/decreasing and their relative maxima/minima, if they exist. Use methods described in class.
- $f(x) = x - x^3$
- $f(x) = x^4 - 4x^3 + 10$
- $f(x) = \dfrac{x + 1}{x}$
- $f(x) = \dfrac{x^2}{x + 1}$
- $f(x) = x\sqrt{x + 1}$

- The height of a rocket in meters follows the polynomial function \[f(t) = -\dfrac{1}{3}t^3 + 16t^2 + 33t + 10\] where $t$ represents $t$ seconds from launch. Determine which seconds the rocket is rising and falling.