Calculus 6.
4 Accumulation Functions Notes
Write your questions
and thoughts here!
Last lesson, we learned about the definite integral. 𝑓 𝑥 𝑑𝑥 represents the area under the
curve of 𝑓 𝑥 on the interval 𝑎, 𝑏 .
Let us say we know the interval starts at 𝑎, but we do not know where it stops. That would
give us 𝒂, 𝒙 where 𝑎 is a constant and 𝑥 is some unknown variable. We can represent that
as a new function that looks like this:
𝐹 𝑥 𝑓 𝑡 𝑑𝑡
1. Let 𝐹 𝑥 𝑓 𝑡 𝑑𝑡. Use the graph of 𝑓 in the figure y
to find the values of the table on the interval 0 𝑥 5.
a) Complete the table.
𝒙 0 1 2 3 4 5
𝑭 𝒙
This is called an ________________________
Fundamental Theorem of Calculus
If 𝑎 is a constant and 𝑓 is a continuous function, then
𝑑
𝑓 𝑡 𝑑𝑡 𝒇 𝒙
𝑑𝑥
Derivatives and Integrals are __________ of each other. They cancel each other out, just like
multiplication and division. In example 1, the graph of 𝑓 is the derivative of 𝐹 𝑥 . So 𝐹 𝑥
is considered the _________________ of 𝑓 𝑥 .
Variations of the FTC
If 𝑎 is a constant, 𝑓 is a continuous function, and 𝑔 and ℎ are differentiable then
𝑑
𝑓 𝑡 𝑑𝑡 𝒇 𝒈 𝒙 ⋅𝒈 𝒙
𝑑𝑥
𝑑
𝑓 𝑡 𝑑𝑡 𝒇 𝒈 𝒙 ⋅𝒈 𝒙 𝒇 𝒉 𝒙 ⋅𝒉 𝒙
𝑑𝑥
Write your questions Find 𝑭 𝒙 .
and thoughts here!
1. 𝐹 𝑥 3𝑡 4𝑡 𝑑𝑡 2. 𝐹 𝑥 sin 𝑡 𝑑𝑡 3. 𝐹 𝑥 ℎ 𝑡 𝑑𝑡
4. 𝐹 𝑥 5𝑡 𝑑𝑡 5. 𝐹 𝑥 𝑡 𝑡 𝑑𝑡
6.4 Accumulation Functions
Calculus
Practice
Find 𝑭 𝒙 .
1. 𝐹 𝑥 𝑡 𝑑𝑡 2. 𝐹 𝑥 5 𝑑𝑡 3. 𝐹 𝑥 4𝑡 𝑡 𝑑𝑡
4. 𝐹 𝑥 cos 𝑡 𝑑𝑡 5. 𝐹 𝑥 𝑡 𝑑𝑡 6. 𝐹 𝑥 sin 𝑡 𝑑𝑡
7. 𝐹 𝑥 𝑑𝑡 8. 𝐹 𝑥 3√𝑡 𝑑𝑡 9. 𝐹 𝑥 2𝑡 𝑑𝑡
10. 𝐹 𝑥 𝑡 𝑑𝑡 11. 𝐹 𝑥 tan 𝑡 𝑑𝑡 12. 𝐹 𝑥 sec 𝑡 𝑑𝑡
13. 𝐹 𝑥 𝑓 𝑡 𝑑𝑡 14. 𝐹 𝑥 4𝑡 1 𝑑𝑡 15. 𝐹 𝑥 3𝑡 1 𝑑𝑡
16. 𝐹 𝑥 𝑡 𝑑𝑡 17. 𝐹 𝑥 𝑡 𝑑𝑡
6.4 Accumulation Functions Test Prep
18. Let 𝑔 𝑥 √𝑡 9 𝑑𝑡. What is 𝑔 4 ?
(A) 5 (B) 3 (C) 3 (D) 4 (E) 5
19. y
x
The graph of a function 𝑓 on the closed interval 0, 6 is shown above. Let ℎ 𝑥 𝑓 𝑡 𝑑𝑡 for 0 𝑥 6.
Find ℎ′ 3 .
(A) 2 (B) 0
(C) 2 (D) Does not exist
20.
𝑓 𝑥
The figure above shows the region 𝐴, which is bounded by the 𝑥- and 𝑦-axes, the graph of 𝑓 𝑥
for 𝑥 0, and the vertical line 𝑥 𝑏. If 𝑏 increases at a rate of units per second, how fast is the area of
region 𝐴 increasing when 𝑏 ?
