Math 109 – Mock Final Solutions– Fall 2019

The Final Exam is cumulative from Chapter 1-6, and students are expected to know the following topics: 1. Marginal Cost, Total Cost Functions, and their applications.

2. Find Definite Integral Using Fundamental Theorem of Calculus (section 6.3).

3. Find Indefinite Integral/Antiderivatives (Section 6.2).

4. Indefinite Integral v.s. Definite Integral.

5. Revenue function, fixed cost, total cost functions, profit functions, break-even point, how to maximize the profit, how to minimize the total cost.

6. Analyzing antiderivatives graphically (section 6.1).

7. Elasticity of Demand (section 4.6).

8. Average Cost (section 4.5) – how to minimize average cost.

9. Exponential growth and decay function (section 1.7).

10. Find area under the curve by approximation (section 5.2).

11. The definite integral as area (section 5.3).

12. The chain rule (section 3.3).

13. Interpretation of the definite integral (section 5.4).

14. Average Value (section 5.6).

15. Total Change and the Fundamental Theorem of Calculus (section 5.5).

16. Present v.s. Future Values (section 6.5).

17. The consumer and producer surplus (section 6.4).

18. Review textbook examples of section 6.3+6.4+6.5 carefully.

1. A company is manufacturing a new dietary supplement pill that promotes body fat reduction. a) The company will supply 60 pills for the price of $40, and will supply 100 pills

for a price of $60. Determine a linear supply function, (q)p S= , for this product.

b) The demand for this new dietary supplement pill is 50 pills at a price of $47.50 and 80 pills at a price of $32.50. Determine a linear demand function, ( )p D q= ,

for this product.

c) Find the equilibrium price and quantity for the new dietary supplement pill.

2. Joanne Wendelken sells silk-screened T-shirts at community festivals and crafts fairs. Her marginal cost to produce on T-shirt is $3.50. Her total cost to produce 60 T-shirts is $300, and she sells them for $9 each. a) Find the linear cost function for Joanne’s T-shirt production.

b) How many T-shirts must she produce and sell in order to break even?

c) How many T-shirts must she produce and sell to make a profit of $500?

3. Lauren Snowden puts $10,500 into an account to save money to buy a car in 12 years. She expects the car of her dreams to cost $35,000 by then. Find the interest rate that is necessary if the interest is computed using the following method: compounded monthly.

4. The cost (in dollars) for manufacturing a particular DVD is ( ) 15,000 6C x x= + ,

where x is the number of DVDs produced. Recall that the average cost per DVD,

denoted by ( )C x , is found by dividing C(x) by x. Find and interpret lim (x)x

C→

5. Find the derivative of the function 4

14 12( ) 2f t

t t= + +

6. If the total revenue received from the sale of x items is given by ( ) 30ln(2 1)R x x= + ,

while the total cost to produce x items is ( )2

xC x = , find the following:

a) The marginal revenue

b) The profit function P(x)

c) The marginal profit when x=60

d) Interpret the results of part (c).

7. Find the point of diminishing returns (x,y) for the given functions, where R(x), represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars).

3 24( ) ( 66 1050 400),0 x 25

27R x x x x= − + + −

8. If the price charged for a candy bar is p(x) cents, then x thousand candy bars will be

sold in a certain city, where ( ) 16010

xp x = − .

a) Find an expression for the total revenue from the sale of x thousand candy bars.

b) Find the value of x that leads to maximum revenue.

c) Find the maximum revenue.

9. The Valve Corporation, a software entertainment company, recently ran a holiday sale on its popular Steam software program. Using a data collected from the sale, it is possible to estimate the demand corresponding to various discounts in the price of the software. Assuming that the original price was $40, the demand for the

software can be estimated by the function 2.8263,751,000q p−= , where p is the price

and q is the demand. Calculate and interpret the elasticity of demand.

10. Find the following 2 3(v e )v dv−

11. Evaluate the definite integral 9

4

(4 3 )r r r dr−

12. A stock analyst plots the price per share of a certain common stock as a function of

time and finds that it can be approximated by the function 0.03(t) 37 6e tS −= + , where t

is the time (in years) since the stock was purchased. Find the average price of the stock over the first six years.

13. ( ) 500f t = represents the rate of flow of money in dollars per year. Assume a 10-

year period at 8% compounded continuously and find the following:

a) The present value

b) The accumulated amount of money flow at 10t =