Show each step of your work and fully simplify each expression.
Turn in your answers in class on a physical piece of paper.
Staple multiple sheets together.
Feel free to use Desmos for graphing.
Answer the following:
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 $[0, 3]$.
Given the function \[f(x) = (x-1)^2(x+1)^2\] state the intervals on which it is increasing and decreasing.
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 $[0, \infty)$.
The total daily cost to manufacture $x$ plastic forks is \[C(x) = 2000 + 2x - 0.0001x^2 \qquad 0 \leq x \leq 10000\]
What is the actual cost in manufacturing the 2001th fork?
What is the marginal cost in manufacturing when $x = 2000$? Interpret the marginal cost in English.
Find the average cost function $\bar{C}$.
Find the marginal average cost function $\bar{C}'$.
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.
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.
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.
Find the average cost function $\bar{C}$.
Find the marginal average cost function $\bar{C}'$. Using the graph in part (a), explain why $\bar{C}'$ is negative for all values of $x$.
Find $\lim_{x\rightarrow\infty}\bar{C}$ and interpret your results.
Elektro plans to open an Ancient Relic item shop. He determined the demand for these items is \[p = -0.04x + 800 \qquad 0 \leq x \leq 20000\]
Find the revenue function $R$.
Find the marginal revenue function $R'$.
Compute $R'(4000)$ and interpret your results.
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.
Find the revenue function $R$ and the profit function $P$.
Find the marginal cost function $C'$, marginal revenue function $R'$ and the marginal profit function $P'$.
Compute $C'(2000), R'(2000)$ and $P'(2000)$ and interpret each number in English.
Suppose we have a demand equation \[p = -0.4x +20\]
Find the elasticity of demand.
Is the demand elastic, unitary, or inelastic at $p = 10$?
Determine the range of values in which demand is elastic, unitary, or inelastic.
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.
Find the elasticity of demand.
When is the demand unitary? (Hint: solve for $E(p) = 1$)
Is the demand elastic or inelastic at $p = 40$? Interpret $E(40)$ in English.
If the unit price is increased slightly from $40$, does the revenue increase or decrease?
A demand equation $x = f(p)$. Keeping in mind problem 6, what happens to the quantity demanded $x$ when you increase the price?
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?
Find the first and second derivatives of the following functions:
$f(u) = \sqrt{4 - 3u}$
$g(a) = a^2(3a+1)^4$
You stood on the edge of a building 225 feet tall and dropped a ball. The distance (in feet) the ball falls in $t$ seconds is \[f(t) = 15t^2\]
At what time does the ball hit the ground?
What is the ball's velocity when it hits the ground?
What is the ball's acceleration when it hits the ground?
The Consumer Price Index for some economy is predicted to follow the function \[f(t) = -0.0003t^{3}+0.009t^{2}+0.0055t+1.29\]
where $f(t)$ is in dollars in tens of thousands and $t = 0$ is the beginning of year 2023.
What is the inflation rate in the beginning of 2040?
Show the rate at which inflation is growing is slowing down in 2040.
Show the rate at which inflation is growing is speeding up in 2028.
Obesity (measured by BMI) in America is approximated by the function \[f(t) = 0.0004t^3 + 0.0036t^2 + 0.8t + 12 \qquad 0 \leq t \leq 13 \]
where $t = 0$ (in years) is the beginning of 1991 and $f(t)$ is percent of people who are obese.
Show $f(t)$ is an increasing function on $0 \leq t \leq 13$. (Hint: $t$ is positive, is each term in $f'(t)$ positive?)
What are the implications of $f(t)$ being an increasing function?
Show the rate of change of the rate of change of the percent of people deemed obese was positive from 1991 to 2004. What are the implications?