Homework 3

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. 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. $f(x) = \dfrac{\sin^4(x)\cos^3(x)}{x^2 - 4}$
  2. Suppose $(1, f(1))$ is a point on the graph of $f(x)$. What is the equation of the tangent line at $x = 1$?
  3. 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? Hint: $0/0$
    2. What always happens to the $h$ in the denominator?
    3. Suppose the above phenomena in (b) does not happen. What do you think went wrong?
  4. Using the limit definition of the slope of a tangent line at $x = a$, i.e. \[f'(a) = \lim_{h\rightarrow 0} \dfrac{f(a + h) - f(a)}{h}\] 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 $(1, 0)$
  5. 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 $s(t) = 4.9t^2$.
  6. What are the three ways to think about the derivative $f'(x)$?
  7. Using the limit definition of a derivative, i.e. \[f'(x) = \lim_{h\rightarrow 0} \dfrac{f(x + h) - f(x)}{h}\] find the derivative of the following functions.
    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. $f(x) = \dfrac{1}{x}$
  8. Suppose $f(x)$ is differentiable on $\mathbb{R}$. Must $f(x)$ be continuous on $\mathbb{R}$? If not, explain why.
  9. 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.
  10. Suppose the graph of $f(x)$ is Sketch a graph of $f'(x)$.
  11. Draw one graph of a function in which all three cases where a function fails to be differentiable appears.
  12. 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 $\mathbb{R}$.
  13. 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.
  14. Consider the function \[f(x) = x^2 - 1\]
    Using the limit definition of the derivative, find $f'(x)$ and $f''(x)$.

  15. Find the derivative using the Differentiation Rules.

    Hint 1: Remember to observe your input variable and to fully simplify each expression.

    Hint 2: You need to know how to manipulate exponents. See Lecture Note II for definitions.

    1. $f(x) = 199$
    2. $g(t) = \left(\sqrt{30}\right)^3$
    3. $f(x) = x^3 + x^2 + x + 1$
    4. $g(v) = \dfrac{a}{v} + bv + \dfrac{c}{v^2}$
    5. $f(x) = \sqrt{x}(x - 1)$
    6. $f(t) = 3t^{-3/4}$
    7. $f(x) = x^{2.9}$
    8. $f(r) = -9r^{2/3}$
    9. $g(x) = 5x^2 + 6x + 7$
    10. $h(r) = \dfrac{r^5 + 2r^4 + 3r + 1}{r}$
    11. $f(a) = \dfrac{5}{a^4} - \dfrac{2}{\sqrt[3]{a^2}} -\dfrac{1}{a} + 200$
    12. $f(x) = \dfrac{5}{x^4} + \dfrac{\sqrt{x^3}}{3} + x$
    13. $f(x) = 2x\left(3x^2 + 1\right)$
    14. $f(x) = (3x + 1)(4x^3 - 5x^2)$
    15. $f(t) = \dfrac{t - \sqrt{t}}{t^{1/3}}$