Math 117
Practice Exam 3
_______________________________________________
Name
Instructor: Nancy Goodisman
1. For the revenue function R(q)  4000q  2q 2 , we can estimate the revenue when 502
units are produced by finding the revenue at q = 500 and using the differential dR as
an estimate for the change in revenue when the quantity increases by 2.
a. What is R(500)?
b. What value do we need to use for dq?
c. Find the derivative, R (500) .
d. Find dR
e. What is your estimate for R(502) using the relationship R(q  dq)  R(q)  dR ?
f. What is the exact value for R(502) using your calculator?
2. The function p  D(q)  500  2q gives the price at which there will be a demand for
q units of a product by consumers. The function p  S (q)  3q  100 gives the price
at which q units will be supplied by the producers.
a. Find the equilibrium quantity and price.
b. Graph the supply curve and the demand curve showing the equilibrium point,
(q 0, p 0 ) .
c. Shade area representing the Producers’ Surplus on the graph between the line
p  p0 and supply curve from q = 0 to q = q0
d. Give the definite integral that represents the shaded area.
Producers’ Surplus = __________________
e. Evaluate the integral from part d.
3. The marginal production cost at q units is
a. Find the anti-derivative of
dC
 11  e 0.05q .
dq
dC
 11  e 0.05q .
dq
b. Find the equation of the total cost function, C(q), if the fixed cost is $200.
30
c. Find the definite integral
 C (q)dq .
20
30
d. Explain what
 C (q)dq represents.
20
4. Find the average value f (t )  3t 2  2t over the interval [2, 4]. Graph the function and
its average value from 2 to 4.
5. For the lines y = x and y 
a.
11 2 1
x  x,
12
12
Find the coordinates of the points where the lines intersect.
b. Graph the lines and shade the enclosed area.
c. Find the area between the two curves.