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:
What are the three ways to think about the derivative $f'(a)$?
Using the limit definition of a derivative, find the derivative of the following functions.
Do not use shortcuts; you will not receive credit for this problem.
$f(x) = 2x + 3$
$g(x) = x + \sqrt{x}$
$f(x) = x^2 - 1$
$g(x) = \dfrac{1}{\sqrt{x}}$
$f(x) = \dfrac{1}{x}$
Suppose $f(x)$ is differentiable on $\mathbb{R}$. Must $f(x)$ be continuous on $\mathbb{R}$? If not, explain why.
Suppose $f(x)$ is continuous on $\mathbb{R}$. Must $f(x)$ be differentiable on $\mathbb{R}$? If not, draw the graph of a function where it fails to be differentiable.
Suppose the graph of $f(x)$ is
Sketch a graph of $f'(x)$.
Draw one graph of a function in which all three cases where a function fails to be differentiable appears.
The graph of $f$ is given. State with reason the $x$ values where $f$ is not differentiable. You may assume the domain of $f$ is $\mathbb{R}$.
Find the derivative using the Differentiation Rules. Remember to observe your input variable and to fully simplify each expression.