Calculus Applications of Integrals
Name_________________________
Main Idea – Total Change
The integral can be used to find total change for a function that describes a rate of change.
!
!
!
! !" = ! ! − !(!)
Tells the total change between a and b for the function F(x).
Remember that a starting value is necessary to find the exact values on F(x).
Examples:
!
If v(t) is a function for volume and v’(t) is the rate of change of volume, then
!′ ! !" will describe the
!
change in volume from time 2 to 4. We would need to know the volume at time t=2 in order to state the
volume at t=4.
!
!′
!
! !" = ! 4 − !!(2)
If c(t) is the concentration of the product of a chemical reaction and c’(t) is the rate of the reaction, then
!
!′
!
! !" will describe the change in concentration from t = 0 to t = 3.
We would need to know the
concentration at time t=0 in order to state the concentration at t=3.
!
!′
!
! !" = ! 3 − !(0)
If dn/dt is the rate of change for a population, then
! !"
!" will describe the change in population from t =
! !"
5 to t = 8. We would need to know the population at time t=5 in order to state the population at t=8.
! !"
!"
! !"
= ! 8 − !(5)
If c(x) is the cost of producing x units of a commodity and c’(x) is the rate of the cost (called marginal cost),
!""
then
!′ ! !" will describe the increase in cost when production is increased from x = 100 to x =
!""
300 units. We would need to know the cost of producing x = 100 units in order to state the cost of producing
at x=3oo units.
!""
!′
!""
! !" = ! 300 − !(100)
Applications in Economics/Business
Cost: a function for how much it costs a company to produce x units of a product. This is usually in the
form of a polynomial:
C(x) = a + bx + bx2 + cx3 etc.
The ordered pair (40, 8) would mean it cost $8 to produce 40 units.
Note: in the above equation, a is called the overhead for the company (rent, heat, maintenance).
Marginal Cost: The derivative of cost. The values are slopes of cost and describe how much it would cost
to produce 1 more unit. The ordered pair (40, 2) would mean it cost $2 to increase production from 40 to 41
units.
Average Cost: The cost per unit produced. Divide the cost function by x.
Ave = C(x) / x
The ordered pair (40, $8) would mean it cost $8 per item to produce 40 units.
Demand: (also called price function) the price a company can charge per item if it sells x units. This
function uses the function name p(x). This function typically decreases – so as the number sold increases,
the cost drops. The ordered pair (100, 8) would mean 100 units will be sold if the price is $8.
Marginal Demand: the change in price by increasing production by 1 unit. The ordered pair (40, -.25)
would mean the price would have to drop by $0.25 in order to increase sales from 41 to 41 units.
Revenue: a function for the amount earned by the company for selling x units. This is found by multiplying
the demand equation by the units sold. The ordered pair (40, 200) would mean that revenue would be $200
dollars if 40 units are sold.
R(x) = x•p(x)
Marginal revenue: (also called the sales function) The derivative of revenue. Tells the rate of change (slope)
of revenue with respect to the units sold and tells how much the company will earn by producing 1 more
unit. The ordered pair (40, 2) would mean that revenue would increase $2 if production changed from 40 to
41 units.
Profit: a function for the money a company has earned after costs are subtracted. The ordered pair (100,
500) would mean it would have a profit of $500 if 100 units were sold.
Profit = Revenue – Cost
Marginal Profit: The derivative of profit. Tells the rate of change (slope) of profit with respect to the units
sold and tells how much profit the company will make by producing 1 more unit. The ordered pair (20, 2)
would mean that profit would increase by $2 to increase production from 20 to 21 units.
Section 1: Displacement, Velocity, and Acceleration
Q1. The position of a particle is given by the equation
in meters.
s(t) = t3 – 6t2 + 9t + 1 where t is in seconds and s is
1. State the equations for velocity and acceleration and accurately graph all 3 functions on [0, 6] below:
Displacement:
Velocity:
Acceleration:
2. What is the velocity and acceleration at time t = 3 seconds? Is the particle moving forward or backward?
3. When is the particle at rest?
4. When does the particle change direction?
5. Set up an integral to find the total change for position on the interval [0, 3] and explain if it makes sense
based on the displacement graph above.
6. Use your answer to part 5 and the value of s(0) to state the position at t = 3.
7. Set up an integral to find the total change for position on the interval [0, 6] and compare this to the total
distance travelled by the particle on the interval [0, 6]
8. Use your answer to part 7 and the value of s(0) to state the position at t = 6.
9. State any inflection points of displacement and explain what this means in terms of the acceleration.
10. Set up an integral to find the total change for velocity on the interval [1, 3].
11. Use your answer to Part 10 and the value of v(1) to state the velocity at t = 3.
Q2. The position of a particle is given be the equation
is in meters.
s(t) = t3 – 12t2 + 36t – 3 where t is in seconds and s
1. State the equations for velocity and acceleration and accurately graph all 3 functions on [0, 8] below:
Displacement:
Velocity:
Acceleration:
2. What is the velocity and acceleration at time t = 4 seconds? Is the particle moving forward or backward?
3. When is the particle at rest?
4. When does the particle change direction?
5. Set up an integral to find the total change for position on the interval [0, 5] and explain if it makes sense
based on the displacement graph above.
6. Use your answer to part 5 and the value of s(0) to state the position at t = 5.
7. Set up an integral to find the total change for position on the interval [0, 8] and compare this to the total
distance travelled by the particle on the interval [0, 8]
8. Use your answer to part 7 and the value of s(0) to state the position at t = 8.
9. State any inflection points of displacement and explain what this means in terms of the acceleration.
10. Set up an integral to find the total change for velocity on the interval [1, 3].
11. Use your answer to Part 10 and the value of v(1) to state the velocity at t = 3.
Q3 If a ball is thrown vertically upward with a velocity of 80 ft/s then its height after t seconds is
s = 80t – 16t2
1. What is the maximum height reached by the ball (no calculator)?
2. What is the velocity of the ball when it is 96 feet above the ground on its way up?
Q4 If the velocity of a rocket launching vertically is v = 1200 + t4 (units = mph) and the rocket is currently
1,000 feet above the ground, then…
1. How much will the velocity change over the next 4 seconds?
2. What will the velocity be at t=4?
3. At what time will the rocket reach a height of 10,000 feet?
Q5 A car starts moving at time t = 0 and goes faster and faster. Its velocity is shown in the following table
at 4 second intervals. Estimate how far the car travels during the 12 seconds.
T (sec)
V (ft/sec)
0
0
4
4
8
7
12
12
Q6 A tank that holds 5000 gallons of water drains in 40 minutes. The volume of water remaining in the
tank can be described using the formula
!
! = 5000 1 − ! !"
!
1. What is the domain for t?
2. At what rate is the water draining from the tank at time t = 5 minutes?
3. At what time is the water flowing out the fastest?
4. Set up an integral that will calculate the change in volume from time t = 10 to time t = 30 and
solve the integral.
Q7 A car starts comes to a stop twelve seconds after the driver applies the brakes. While brakes are on the
following velocities are recorded. Estimate how far the car travels during the 12 seconds.
T (sec)
V (ft/sec)
0
12
2
11
4
7
6
5
8
4
10
2
12
0
Q8 Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their
velocity functions:
1. Which car is ahead after one minute? Explain.
2. What is the meaning of the area of the shaded region?
3. Which car is ahead after 2 minutes? Explain.
4. Estimate the time at which the cars are again side-by-side.
Q9 A car’s acceleration is described by the function a = 0.5t – 4. If the car’s velocity is 16 m/s at time t = 0
seconds, then find the total distance traveled by the car on [0, 8].
Section 2: Applications in Economics/Business
Q10 The marginal cost for x units is C’(x) = 0.006x2 – 1.5x + 8 (measured in dollars per unit). And the fixed
start-up cost is C(0) = $1,500,000. Find the total change in cost for producing the first 2000 units.
Q11 The marginal revenue from selling x items is 90 – 0.02x. The revenue from the sale of the first 100
items is $8800. What is the revenue from the sale of the first 200 items?
Q12 The marginal cost of production x units of a certain predict is 140 – 0.5x + 0.012x2 (in dollars per unit).
Find the increase in cost if the production level is raised from 3000 units to 5000 units.
Q13 A manufacturer has been selling 1000 television sets a week at $450 each. A survey indicates that for
every $10 that the price is reduced, the number of sets sold will increase by 100 a week. No Calculator for
this question.
1. Find the demand function.
2. Find the revenue function.
3. What should the price be to maximize revenue?
4. What is the increase in revenue if the company sales increase from 1000 units to 1500 units based on the
revenue function above?
Q14 The cost, in dollars of producing x yards of a fabric is C(x) = 2000 + 3x + 0.01x2 + 0.002x3
1. Find C(100) and C(101). How did cost change from 100 to 101 yards?
2. Find the marginal cost function C’(x)
3. Find C’(100), compare this to the answer in part 1, and describe the meaning of C’(100).
4. Set up an integral to calculate the total change in cost when increasing production from 100 to 300 yards.
5. The company used to produce 200 yards and increased production to x yards. If the total change in cost
for the increasing production is $1,000,000 then how much fabric is now being produced?
Q15 Consider the cost function C(x) = 3700 + 5x – 0.04x2 +.0003x3
1. Find the equation for the average cost to produce x items.
2. Find the average cost to produce 100 items.
3. Graph the equation in part 2 on [0, 1000].
4. Find the number of units that should be produced to minimize the cost per unit.
5. Find the minimum value for marginal cost and explain what it means.
Q16 The cost function for a commodity is C(x) = 84 + 0.16x – 0.0006x2 + 0.000003x3
1. Find and interpret C’(100)
2. Compare C’(100) with the cost of producing the 101st item (consider the meaning of marginal cost).
3. Find any inflection points for the cost function and explain their meaning.
4. Set up and solve an integral to calculate the total change for the marginal cost from 10 items to 20 items.
Q17 The cost function for a company is C(x) = 680 + 4x + 0.01x2 and the demand function is
p(x) = 12 – x/500.
1. Find an equation for total profit.
2. Maximize total profit.
3. Find the equation for marginal profit.
4. Find the value of marginal profit for 100 units and explain what it means.
5. Set up an integral to calculate the change in total profit by increasing production from 100 to 200 units.
Q18 Annual coal production in the US (in quadrillion BTU per year) is given in the following table.
Estimate the total amount of coal produced in the US between 1970 and 1990.
Year
Rate of Coal
production
1970
14.61
1975
14.99
1980
18.60
1985
19.33
1990
22.46
Q19 The figure shows the graphs of the marginal revenue function R’ and the marginal cost function C’ for
a manufacturer. R and C are measured in thousands of dollars.
1. Find the marginal revenue for producing 50 items and 100 items.
2. What does it mean to say that R’(x) has a negative slope? Use your answers from part 1 to describe.
3. What does it mean to say that C’(50) is a local minimum?
4. Find the total change in revenue from increasing production from 50 to 100 units.
5. What is the meaning of the area of the shaded region?
Interesting uses for the Fundamental Theorem of Calculus:
Q20 A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $10, the average
attendance had been 27,000. When ticket prices were lowered to $8, the average attendance rose to 33,000.
1. Find the demand function, assuming that it is linear.
2. How should the ticket prices be set to maximize revenue?
Q21 Given that F(1) = 0.5 and f(x) = sin(ln(x)), find F(3)
Q22 Given that F(1) = 2 and f(x) = x2 then find the value of t where F(t) – F(1) = 8.
Q23 Given that the acceleration of a car is described by a(t) = 4t – 12 and that the current velocity of the car
is 20 mph, find how long it will take the car to reach a velocity of 50 mph.