Chapter 13 – 5a The Fundamental Theorem of Calculus
Example 1. The price-demand equation for hamburgers at a fast-food restaurant is given by
𝑥 + 400𝑝 = 2,000
where x is the number of hamburgers per day and p is the price per hamburger. The total cost (in dollars) for
producing x hamburgers is given by
𝐶(𝑥) = 400 + 1.35𝑥
A) Assume that hamburger sales increase from 150 hamburgers per day to 200 hamburgers per day. Compare the
change in total cost with the area under the marginal cost curve from 150 to 200.
B) Under the same assumptions, compare the change in revenue with the area under the marginal revenue curve
from 150 to 200.
C) Under the same assumptions, compare the change in profit with the area under the marginal profit curve from
150 to 200.
The Fundamental Theorem of Calculus
If 𝑓(𝑥) is continuous on the interval [𝑎, 𝑏] and 𝐹(𝑥) is any antiderivative of 𝑓(𝑥) then
𝑏
∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) = 𝐹(𝑥)|𝑏𝑎
𝑎
Average Value of a Continuous Function on [a,b]
If 𝑓(𝑥) is continuous on the interval [𝑎, 𝑏] and 𝐹(𝑥) is any antiderivative of 𝑓(𝑥) then the average value of 𝑓(𝑥) on
the interval [𝑎, 𝑏]is given by
𝑏
1
𝐹(𝑏) − 𝐹(𝑎)
∫ 𝑓(𝑥)𝑑𝑥 =
𝑏−𝑎 𝑎
𝑏−𝑎
3
Example 4. Evaluate ∫2 2𝑥 𝑑𝑥
f(x) = 2x
10
8
6
4
2
0
0
1
2
3
4
3
4
3
Example 5. Evaluate ∫1 2𝑥 − 3 𝑑𝑥
f(x) = 2x - 3
6
4
2
0
-2
-4
0
1
2
3.5
Example 6. Evaluate ∫1 2𝑥 −1 𝑑𝑥
4
3
2
2x^-1
2x^-2
Example 7. Evaluate
3.5
∫1 2𝑥 −2
𝑑𝑥
1
0
0
1
2
3
4
4
Example 8. Evaluate ∫2 (8𝑥 3 − 2𝑥 4 ) 𝑑𝑥
8x3-2x4
60
50
40
30
20
10
0
0
1
2
3
4
10
Example 9. Evaluate ∫−5 𝑒 −0.05𝑥 𝑑𝑥
f(x) = e-0.05x
1.5
1
0.5
0
-6
𝑒 ln 𝑡
Example 10. Evaluate ∫1
𝑡
-4
𝑑𝑡
-2
0
2
4
6
8
10 12
f(x) = (ln t)/t
1
0.5
0
0
-0.5
-1
0.5
1
1.5
2
2.5
3