Final Exam Review
1. The revenue (in dollars) for a product is modeled by 𝑅 = 60𝑒 0.5𝑥 where 𝑥 is the number of units.
a. What is the revenue from 12 units? Round your answer to the nearest cent.
b. How many units must be sold to have revenue of $15,000? Round your answer to the nearest unit.
c. How fast is the revenue changing at 12 units?
d. What is the differential of 𝑅? Use differentials to approximate the change in revenue between 12 and 13
units. Compare this value to the actual change in revenue.
2. Evaluate lim
𝑥→0
𝑥
𝑥 3 +𝑥 2 −2𝑥
3
3. Let for 𝑓(𝑥) = √𝑥 2 − 9
a. Write the equation of the tangent line at x = 4
3
b. Find the critical numbers for 𝑓(𝑥) = √𝑥 2 − 16 and classify each as a relative maximum, relative
minimum or neither.
4. Use the graph of 𝑦 = 𝑓(𝑥) below to answer parts a-c.
a. Fill in the blanks. If an answer does not exist, write DNE
lim 𝑓(𝑥) = _______ , 𝑓 ′ (−10) = ________,
lim 𝑓(𝑥) = _________
𝑥→−10
b. 𝑓 ′ (5) = _______,
𝑥→4
𝑓 ′ (−9) = _______,
𝑓 ′ (−13) = _______
c. Name one value of x where both 𝑓′ and 𝑓′′ are positive.
5. The demand of a product can be modeled by 𝑝 = −𝑥 2 + 5𝑥 + 20 and the supply can be modeled by 𝑝 =
2𝑥 2 + 9𝑥 where 𝑝 is the price per units and 𝑥 is the number of units.
a. What is the demand and supply at level of production of 4 units?
b. At what level of production is the point of equilibrium for this product? What is the price at this
point of equilibrium?
c. Using the demand function 𝑝 = −𝑥 2 + 5𝑥 + 20,
i. what is the revenue from selling 7 units?
ii.
what is the revenue from selling 𝑥 units?
iii.
using your answer in part ii, what is the marginal revenue for 5 units?
d. What is the consumer surplus?
6. Suppose the demand of a product is given by 𝑝 = 950 − 25𝑥 where 𝑝 is the price per unit and 𝑥 is the
number of units. Also, the cost is 𝐶(𝑥) = 50𝑥.
a. What is the revenue for 𝑥 units?
b. What is the profit function?
c. Determine if demand is elastic, inelastic or is of unit elasticity when 𝑥 = 30.
d. What level of production maximizes revenue?
7. Find the particular equation for 𝑓(𝑥) if 𝑓 ′ (𝑥) = 𝑥 2 (2𝑥 − 1) and 𝑓(1) = 7.
1
8. Evaluate ∫0 (𝑥 2 + 2𝑥)2 𝑑𝑥 . You must show your work.
𝑑𝑅
9. The marginal revenue of a company is modeled by 𝑑𝑥 = −5𝑥 + 𝑒 𝑥
a. If 𝑅(0) = 5, what is the particular equation for 𝑅(𝑥)?
b. At what value of 𝑥 is the point of diminishing returns? Round your answer to 1 decimal place.
10. Evaluate ∫
𝑥 3 −𝑥+5
𝑥3
𝑑𝑥
11. Solve each equation.
a. ln(2𝑥 − 9) − ln 𝑥 = 3
b. 4𝑒 .02𝑥 + 9 = 20
c. 𝑥 4 = 9𝑥 7