Homework 7

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. What are the three ways to think about the derivative $f'(a)$?
  2. 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. $f(x) = \dfrac{1}{x}$
  3. Suppose $f(x)$ is differentiable on $\mathbb{R}$. Must $f(x)$ be continuous on $\mathbb{R}$? If not, explain why.
  4. 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.
  5. Suppose the graph of $f(x)$ is Sketch a graph of $f'(x)$.
  6. Draw one graph of a function in which all three cases where a function fails to be differentiable appears.
  7. 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 $\mathbb{R}$.
  8. Find the derivative using the Differentiation Rules. Remember to observe your input variable and to fully simplify each expression.
    1. $f(x) = 199$
    2. $g(t) = \left(\sqrt{30}\right)^3$
    3. $f(x) = x^3 + x^2 + x + 1$
    4. $f(v) = \dfrac{a}{v} + bv + \dfrac{c}{v^2}$
    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. $f(x) = 5x^2 + 6x + 7$
    10. $f(r) = \dfrac{r^5 + 2r^4 + 3r + 1}{r}$
    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. $f(t) = \dfrac{t - \sqrt{t}}{t^{1/3}}$
    16. $y = x + \dfrac{1}{x}$
    17. $f(x) = (x + 1)(x^{-2} + x^{-3})$
    18. $g(x) = 4\sin x - 5 \cos x$
    19. $f(\theta) = 13424e^\theta$
    20. $f(x) = 3x^5 - 4e^x - \sin x + \cos x$
  9. Find the equation of the tangent line for the function $f(x) = \cos x$ at the point $(0,1)$.
  10. Find the 51st derivative of $f(x) = e^x + \cos x$.
  11. The number of tree species $S(a)$ in a given area $a$ in the Pasoh Forest Reserve in Malaysia has been modeled by the power function \[S(a) = 0.882a^{0.842}\] where $a$ is measured in square meters. Find $S'(100)$ and interpret the answer in English.
    Source: K. Kochummen et al., "Floristic Composition of Pasoh Forest Reserve, a Lowland Rain Forest in Peninsular Malaysia," Journal of Tropical Forest Science 3