Homework 4

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.

Answer the following:

  1. Use the three-part definition of continuity to explain why the function is continuous at the given number $a$.
    1. $\displaystyle f(x) = \begin{cases}(x-1)^2& x < 0 \\ (x+1)^2 & x \geq 0\end{cases} \qquad a = 0$
    2. $\displaystyle f(x) = \begin{cases}x(x-1)& x < 1 \\ 0 & x = 1 \\ \sqrt{x-1} & x > 1\end{cases} \qquad a = 1$
  2. Use the three-part definition of continuity to explain why the function is discontinuous at the given number $a$. Sketch the graph of the function.
    1. $\displaystyle f(x) = \begin{cases}x + 1 & x < 0 \\ x^2 & x \geq 0\end{cases} \qquad a = 0$
    2. $\displaystyle f(x) = \begin{cases}x + 5 & x < 0 \\ 2 & x = 0 \\ -x^2 + 5 & x > 0\end{cases} \qquad a = 0$
    3. $\displaystyle f(x) = \begin{cases}-x & x < 0 \\ 1 & x = 0 \\ x & x > 0\end{cases} \qquad a = 0$
  3. If a function $f(x)$ is continuous at $x = a$, what does $\displaystyle \lim_{x\rightarrow a} f(x)$ have to be?
  4. How would you define $f(2)$ in the function \[f(x) = \dfrac{x^2 - x - 2}{x - 2}\] in order to make $f(x)$ continuous at $x = 2$?
    Hint: find the hole in $f(x)$ and "fill in" the hole.
  5. State in interval notation where each of the following functions are continuous.
    1. $f(x) = \dfrac{\sin(x)}{x^2}$
    2. $f(x) = 4x^{32} - 8x^2 + x - 1$
    3. $f(x) = x^{32894983} - 2x + \dfrac{1}{x}$
    4. $f(x) = \dfrac{\sin^4(x)\cos^3(x)}{x^2 - 4}$
  6. Use the continuity limit swap theorem to prove the following function is continuous at 4. \[\displaystyle f(x) = \begin{cases} \sqrt{x} & x < 4 \\ 2\cos\left(x-4\right) & x \geq 4\end{cases} \qquad a = 4\]
  7. Draw three different graphs, each of which shows one of each of a removeable, jump, and infinite discontinuity.
  8. Draw one graph of a function which satisfies the following conditions simultaneously:
    • Jump discontinuity at $2$ but continuous from the right at $2$
    • Discontinuous at $-1$ and $4$ but continuous from the left at $-1$ and from the right at $4$.
    • Continuous everywhere else
  9. Show $f(x) = x^4 + x - 3$ has a root between $(1, 2)$.
  10. Suppose $(1, f(1))$ is a point on the graph of $f(x)$. What is the equation of the tangent line at $x = 1$?
  11. When we are using the definition of the slope of the tangent line at the point $(a, f(a))$:
    1. What type of limit is this called?
    2. What always happens to the $h$ in the denominator?
    3. Suppose the above phenomena does not happen. What do you think went wrong?
  12. 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 an equation of the tangent line for the following functions at the given point:
    1. $f(x) = 4x - 3x^2$ at the point $(2, -4)$
    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 $(1, 0)$

    13. The rest of these problems will appear on next week's homework. Skip the rest of these for Homework 4.

  13. 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 $s(t) = 4.9t^2$.
  14. What are the three ways to think about the derivative $f'(a)$?
  15. Using the limit definition of a derivative, find the derivative of the following functions at $x = a$.
    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. $f(x) = \dfrac{1}{x}$
  16. Suppose $f(x)$ is differentiable on $\mathbb{R}$. Must $f(x)$ be continuous on $\mathbb{R}$? If not, explain why.
  17. 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.
  18. Suppose the graph of $f(x)$ is Sketch a graph of $f'(x)$.
  19. Draw one graph of a function in which all three cases where a function fails to be differentiable appears.
  20. 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 $\mathbb{R}$.