Math · Calculus I · Worked example
Find an accumulation function and its slope
Let F(x) = ∫₀ˣ (t − 1) dt. Find a formula for F and its derivative, and explain where F decreases and why F(2) = 0.
Find an antiderivative of the integrand
t²/2 − t has derivative t − 1.
Subtract the endpoint values
By Part 2, F(x) = G(x) − G(0), and G(0) = 0.
Read the slope
Part 1 says F′(x) is the integrand evaluated at x, so F′(x) = x − 1. Differentiating the formula agrees.
Where F decreases
F′(x) = x − 1 is negative for x < 1. On [0, 1] the integrand is below the axis, so each new strip subtracts area and F falls, reaching −1/2 at x = 1. After that the strips add.
Why F(2) = 0
From 0 to 1 the signed area is −1/2, a triangle below the axis; from 1 to 2 it is +1/2. They cancel, so F(2) = 0 even though the total area is 1.
Result
F(x) = x²/2 − x and F′(x) = x − 1. F decreases for x < 1 and increases for x > 1, and F(2) = 0 because the areas below and above the axis cancel.
Your turn
Find the slope of F(x) = ∫₁ˣ t² dt at x = 3.
Show the answer and explanation
The slope is 9.
By Part 1, F′(x) = x², so F′(3) = 9. There is no need to integrate first; doing so gives F(x) = x³/3 − 1/3, whose derivative is the same x².
Keep exploring
In Accumulation / FTC, move x from 0 to 2. F falls to −1/2 at x = 1, where its tangent is flat, then climbs back to 0 at x = 2.
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