Finding the values for A(x) for domain 0≤t≤8 when A(x)∫ f(t)dt w/ interval [0,x]

Question

How to find: A(0), A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8) using A(x) = ∫^x₀ f(t)dt on the domain 0 ≤ t ≤ 8.

The graph points are: (0,-2), (1,-2), (2,-4), (3,0), (4,1), (5,2), (6,2), (7,0), (8,0).

I also need to find the intervals where A(x) is increasing and decreasing for x.

Lastly, I also need to need the maximum and minimum values for A(x) when x=?

I've been stuck on his question for hours and do not know what I am doing wrong... I've been trying to use the fundamental theorem of calculus...

Answer 1

I briefly explain each of the concepts you're being asked to understand in this video. I couldn't go into detail with the nuances and such due to hitting the max video length, but I'd be happy to answer any questions or such in this comment thread or in my direct messages! Hope this helps!

Answer 2

Given Points:

(0,−2),(1,−2),(2,−4),(3,0),(4,1),(5,2),(6,2),(7,0),(8,0).

Area between each interval by multiplying the width by the height (i.e., Δx×f(t) )starting from x=0up to x=8

· Find Intervals of Increase/Decrease:

• f(t) to determine when A(x) is increasing when f(t)>0or decreasing when f(t)<0

• Increasing intervals are x=4,6 and decreasing intervals are x=0,3

Increasing: x=4

Decreasing: x=0

· Maximum and Minimum Values:

From the computed values of A(x) is 1 at x = (6,7,8) and A(x) is -4 at x= (2,3), find where the function attains its maximum and minimum.

Maximum Value: 1

Minimum Value: -4