TUTORIAL 8-9-10-11
1. Determine the following:
a)
 (x
3
 6 x 2  x)dx
b)
t
c)  (
 4(t  3)  2 )dt
4
2. Which of the following is
d)
x
a)
2
2
( x  1) 5 / 2  ( x  1) 3 / 2  C
5
3
b)
1 2 2
x * ( x  1) 3 / 2  C
2
3
(
2
x
 3 x )dx
 (4  5e
5t
e 2t

)dt
3
x  1dx ?
3. Let P(t) be the total output of a factory assembly line after t hours of work. Suppose that
the rate of production at time t is 60+2t-0.25t2 units per hour. Find the formula for P(t).
[Hint: The rate of production is P’(t) and P(0) = 0.]
4. A flu epidemic hits a town. Let P(t) be the number of persons sick with the flu at time t,
where time is measure in days from the beginning of the epidemic and P(0) = 100. Suppose
that after t days the flu is spreading at the rate of 120t – 3t2 people per day. Find the
formula for P(t).
5. Suppose the drilling of an oil well has a fixed cost of $10,000 and a marginal cost of C’(x)
= 1000 + 50x $/m. Find the expression for C(x), the total cost of drilling x meters.
6. Use a Riemann sum with n = 4 and left endpoints to estimate the area under the graph
of f(x) = 4-x on the interval [1,4]. Then repeat with n = 4 and midpoints. Compare the
answer with the exact answer, 4.5, which can be computed from the formula for the area of
a triangle.
7. The velocity of a car is recorded from the speedometer every 10 seconds, beginning 5
seconds after the car starts to move (see table below). Use a Riemann sum to estimate the
distance the car travels during the first 60 seconds. [Note: Each velocity is given at the
middle of a ten-second interval. The first interval extends from 0 to 10, and so on.]
A Car’s Velocity
Time
Velocity
5
15
25
35
45
55
20
44
32
39
65
80
8. Calculate the following definite integrals
3
6
a)
1
 x dx
b)
3
x / 10
2
 (e  x  1)dx
5
dt
0
d)

25  3t dt
3
5
e)
3
1
1
4
c) 
dt
3
1 (t  2)
 (5t  1)
4
f)
 (x
2
2

2
1

dx
2
x5
x
9. The worldwide rate of cigarette consumption (in trillions of cigarettes per year) since
1960 is given approximately by the function c(t) = 0.1t + 2.4, where t = 0 corresponds to
1960. Determine the number of cigarettes sold from 1980 to 2004.
10. Suppose that the marginal profit function for a company is 100 + 50x – 3x2 at
production level x.
a) Find the extra profit earned from the sale of 3 additional units if 5 units are currently
being produced.
b) Describe the answer to part (a) as an area. (Do not make a sketch).
11. The marginal profit for a certain company is MP1(x) = -x2 + 14x – 24. The company
expects the daily production level to rise from x = 6 to x = 8 units. The management is
considering a plan that would have the effect of changing the marginal profit to MP2(x) = x2 + 12x – 20. Should the company adopt the plan? Determine the area between the graphs
of the two marginal profit functions from x = 6 to x = 8. Interpret this area in economic
terms.
12. Find the area of the region bounded by y = 1/x2, y = x and y = 8x for x ≥ 0.
13. Assuming that a country’s population is now 3 million and is growing exponentially with
growth constant 0.02, what will be the average population during the next 50 years?
14. One hundred dollars are deposited in the bank at 5% interest compounded
continuously. What will be the average value of the money in the account during the next
20 years?
15. Determine the point of intersection (Q, P) and the consumers’ and producers’ surplus
given demand curve P =
25  0.1x and supply curve
0.1x  9 – 2.
16. Suppose that money is deposited daily into a savings account at an annual rate of
$2000. If the account pays 5% interest compounded continuously, estimate the balance in
the account at the end of 3 years.
17. An investment pays 10% interest compounded continuously. If money is invested
steadily at the rate of $5000 per year, how much time is required until the value of the
investment reaches $140,000?
18. Suppose that in a certain country the daily demand for coffee is given by f(p1,p2) =
16p1/p2 thousand kg, where p1 and p2 are the respective prices of tea and coffee in dollars
per kg. Compute and interpret f(3,4).
19. In c certain suburban community, commuters have the choice of getting into the city by
bus or train. The demand for these modes of transportation varies with their cost. Let
f(p1,p2) be the number of people who will take the bus when p1 is the price of the bus ride
and p2 is the price of the train ride. Explain why the partial derivative with respect to p1
and p2 are less than 0 and greater than 0 respectively.
20. Using data collected from 1929-1941, Richard Stone determined that the yearly
quantity Q of beer consumed in the United Kingdom was approximately given by the
formula Q = f(m,p,r,s) where
f(m,p,r,s) = (1.058)m0.136p-0.727r0.914s0.816
and m is the aggregate real income (personal income after direct taxes, adjusted for retail
price changes), p is the average retail price of the commodity (in this case, beer), r is the
average retail price level of all other consumer goods and services, and s is a measure of
the strength of the beer. Determine which partial derivatives are positive and which are
negative and give interpretations.
21. A company manufactures and sells two products, call them I and II, that sell for $10
and $9 per unit, respectively. The cost of producing x units of product I and y units of
product II is 400 + 2x + 3y + 0.01(3x2 + xy + 3y2).
Find the values of x and y that maximize the company’s profits.
22. Four hundred eighty dollars are available to fence in a rectangular garden. The fencing
for the north and south sides of the garden costs $10 per meter and the fencing for the east
and west sides costs $15 per meter. Find the dimensions of the largest possible garden.
23. A firm makes x units of product A and y units of product B and has a production
possibilities curve given by the equation 4x2 + 25y2 = 50,000 for x ≥ 0, y ≥ 0. Suppose
profits are $2 per unit for product A and $10 per unit for product B. Find the production
schedule that maximizes the total profit.