Homework 5


  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. Draw two functions, one of which is increasing on $\mathbb{R}$ and the other which is decreasing on $(-\infty, 0] \cup [3, \infty]$ and increasing on $[0, 3]$.
  2. Given the function \[f(x) = (x-1)^2(x+1)^2\] state the intervals on which it is increasing and decreasing.
  3. The total cost $C(x)$, as we know, is a function that outputs the total cost to produce $x$ units. Explain why $C(x)$ is an increasing function on $[0, \infty)$.
  4. The total daily cost to manufacture $x$ plastic forks is \[C(x) = 2000 + 2x - 0.0001x^2 \qquad 0 \leq x \leq 10000\]
    1. What is the actual cost in manufacturing the 2001th fork?
    2. What is the marginal cost in manufacturing when $x = 2000$? Interpret the marginal cost in English.
    3. Find the average cost function $\bar{C}$.
    4. Find the marginal average cost function $\bar{C}'$.
  5. Suppose the marginal profit function is positive at $x = a$. Should we decrease the level of production? Give at least one reason to support your claim.
  6. Suppose you are given a demand function $p = f(x)$ where $x$ is quantity demanded. You manage to solve for $x$, so you now have an equation $x = f(p)$. Explain in economic terms why $f(p)$ is a decreasing function.
  7. Costco decided to open a new factory to increase rotisserie chicken output. It is estimated the total cost for producing $x$ rotisserie chickens is \[C(x) = 5.2x + 200000\] dollars per year.
    1. Find the average cost function $\bar{C}$.
    2. Find the marginal average cost function $\bar{C}'$. Using the graph in part (a), explain why $\bar{C}'$ is negative for all values of $x$.
    3. Find $\lim_{x\rightarrow\infty}\bar{C}$ and interpret your results.
  8. Elektro plans to open an Ancient Relic item shop. He determined the demand for these items is \[p = -0.04x + 800 \qquad 0 \leq x \leq 20000\]
    1. Find the revenue function $R$.
    2. Find the marginal revenue function $R'$.
    3. Compute $R'(4000)$ and interpret your results.
  9. The weekly demand for Liv's Legendary Items is \[p = 600 - 0.05x \qquad 0 \leq x \leq 12000\] where $p$ is the unit price in dollars and $x$ is the quantity demanded. The weekly total cost associated with manufacturing Liv's Legendary Items is given by \[C(x) = 0.000002x^3 - 0.03x^2 + 400x + 80000\] where $x$ is the cost in producing $x$ items.
    1. Find the revenue function $R$ and the profit function $P$.
    2. Find the marginal cost function $C'$, marginal revenue function $R'$ and the marginal profit function $P'$.
    3. Compute $C'(2000), R'(2000)$ and $P'(2000)$ and interpret each number in English.
  10. Suppose we have a demand equation \[p = -0.4x +20\]
    1. Find the elasticity of demand.
    2. Is the demand elastic, unitary, or inelastic at $p = 10$?
    3. Determine the range of values in which demand is elastic, unitary, or inelastic.
  11. Suppose we have a demand equation \[x = \sqrt{400 - 5p} \qquad 0 \leq p \leq 80\] where $p$ is in dollars and the quantity demanded $x$ is in hundreds.
    1. Find the elasticity of demand.
    2. When is the demand unitary? (Hint: solve for $E(p) = 1$)
    3. Is the demand elastic or inelastic at $p = 40$? Interpret $E(40)$ in English.
    4. If the unit price is increased slightly from $40$, does the revenue increase or decrease?
  12. A demand equation $x = f(p)$. Keeping in mind problem 6, what happens to the quantity demanded $x$ when you increase the price?
  13. Suppose we have a demand equation $x = f(p)$ and it is inelastic at $p = a$. Keeping in mind the previous question, why does increasing the price increase the revenue?
  14. Find the first and second derivatives of the following functions:
    1. $f(u) = \sqrt{4 - 3u}$
    2. $g(a) = a^2(3a+1)^4$