Grade 12 · Math · Calculus

12th grade The Fundamental Theorem of Calculus worksheet

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The Fundamental Theorem of Calculus

Grade 12 Math Worksheet

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Esme evaluates ∫ from 1 to 2 of (3x² + 2x) dx with an antiderivative. What is it?
Kira evaluates ∫ from 3 to 4 of (3x² + 2x) dx with an antiderivative. What is it?
Luz evaluates ∫ from 1 to 5 of (3x² + 2x) dx with an antiderivative. What is it?
Milo defines g(x) = ∫ from 1 to x² of (t + 3) dt. What is g′(2)?
Ren defines g(x) = ∫ from 1 to x² of (t + 5) dt. What is g′(2)?
Zee defines g(x) = ∫ from 1 to x² of (t + 6) dt. What is g′(4)?
7.
Tala defines g(x) = ∫ from 0 to x of f(t) dt, with f shown. Where on [0, 7] does g have its largest value?
- x = 1, where the graph of f is at its highest
- x = 3, where f changes from + to −
- x = 7, the right end
8.
Pia defines g(x) = ∫ from 0 to x of f(t) dt, with f shown. Where on [0, 9] does g have its smallest value?
- x = 4, where f changes from − to +
- x = 9, the right end
- x = 2, where the graph of f is at its lowest
9.
Bo defines g(x) = ∫ from 0 to x of f(t) dt, with f shown. Where on [0, 6] does g have its largest value?
- x = 2, where the graph of f is at its highest
- x = 6, the right end
- x = 5, where f changes from + to −
10.
Ari defines g(x) = ∫ from 0 to x of f(t) dt, with f shown. Where on [0, 8] does g have its smallest value?
- x = 2, where f changes from − to +
- x = 8, the right end
- x = 1, where the graph of f is at its lowest

The Fundamental Theorem of Calculus, both parts. Every version has 10 questions, and the answer key prints on the second page.

I can find g′(x) for g(x) = ∫ from a to x of f(t) dt, including x² and x at the bottom.
I can evaluate a definite integral as F(b) − F(a).
I can say where an accumulation is largest or smallest from the sign of f.
I can find an amount from its start and the integral of its rate.

Standard: none (beyond CCSS)

accumulation function: g(x) = ∫ from a to x of f(t) dt: the area so far.

antiderivative: A function F with F′ = f.

moving end: The end of the integral that is x, not a number.

Subtracted f(b) − f(a) using the function itself. Part 2 needs an antiderivative F with F′ = f.

Added F(b) and F(a). The theorem takes the top minus the bottom.

Took the derivative of f. The derivative of the accumulation is f itself.

What is the most common mistake with the fundamental theorem of calculus?

Subtracted f(b) − f(a) using the function itself. Part 2 needs an antiderivative F with F′ = f.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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