Homework 5

Directions:

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

Note: if the problem says to use the limit definition and you use the shortcut, you will receive zero credit for that problem.

Answer the following:

  1. A ball is dropped from a tower 1000 meters tall. Find the instantaneous velocity after 4 seconds.
    Hint: Use Galileo's Law, which says the total displacement for a freely falling body after $t$ seconds is described by the function $s(t) = 4.9t^2$.
  2. Using the limit definition of the slope of a tangent line, i.e. \[m = \lim_{h\rightarrow 0} \dfrac{f(a + h) - f(a)}{h}\] find the equation of the tangent line for the following functions at the given point:
    1. $f(x) = x^2 - 1$ at the point $(1, 0)$
    2. $f(x) = \dfrac{2x}{x + 1}$ at the point $(1, 1)$
    3. $f(x) = \sqrt{x}$ at the point $(1, 1)$
    4. $f(x) = (x-1)^2$ at the point $(1, 0)$
  3. What are the three ways to think about the derivative $f'(a)$?
  4. 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.
    1. $f(x) = 2x + 3$
    2. $g(x) = x + \sqrt{x}$
    3. $f(x) = x^2 - 1$
    4. $g(x) = \dfrac{1}{\sqrt{x}}$
    5. $f(x) = \dfrac{1}{x}$
  5. Suppose $f(x)$ is differentiable on $\mathbb{R}$. Must $f(x)$ be continuous on $\mathbb{R}$? If not, explain why.
  6. 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.
  7. Suppose the graph of $f(x)$ is Sketch a graph of $f'(x)$.
  8. Draw one graph of a function in which all three cases where a function fails to be differentiable appears.
  9. 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 $\mathbb{R}$.

    10. The rest of these problems will appear on next week's homework. Skip the rest for this homework.

  10. Consider Galileo's Law for freely falling bodies: \[s(t) = 4.9t^2\] Using the limit definition of the derivative, find the velocity $v(t)$ and acceleration $a(t)$ functions.
  11. Consider the function $f(x) = x^2 - 1$.
    Using the limit definition of the derivative, find $f'(x)$ and $f''(x)$.