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  2. AP Calculus
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Explain FTOC Part 1.

The derivative of the definite integral of a function with respect to its upper limit is the original function evaluated at that upper limit.

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Explain FTOC Part 1.

The derivative of the definite integral of a function with respect to its upper limit is the original function evaluated at that upper limit.

Explain FTOC Part 2.

The definite integral of a function from a to b is the difference between the antiderivative evaluated at b and the antiderivative evaluated at a.

What is the relationship between differentiation and integration according to the FTOC?

Differentiation and integration are inverse operations; one can 'undo' the other.

What does the Fundamental Theorem of Calculus, Part 1 state?

If g(x)=∫axf(t)dtg(x) = \int_{a}^{x} f(t) dtg(x)=∫ax​f(t)dt, then g′(x)=f(x)g'(x) = f(x)g′(x)=f(x).

What does the Fundamental Theorem of Calculus, Part 2 state?

If F(x)F(x)F(x) is an antiderivative of f(x)f(x)f(x), then ∫abf(x)dx=F(b)−F(a)\int_{a}^{b} f(x) dx = F(b) - F(a)∫ab​f(x)dx=F(b)−F(a).

What is the formula for FTOC Part 1?

g(x)=∫axf(t)dt  ⟹  g′(x)=f(x)g(x) = \int_{a}^{x} f(t) dt \implies g'(x) = f(x)g(x)=∫ax​f(t)dt⟹g′(x)=f(x)

What is the formula for FTOC Part 2?

∫abf(x)dx=F(b)−F(a)\int_{a}^{b} f(x) dx = F(b) - F(a)∫ab​f(x)dx=F(b)−F(a), where F is the antiderivative of f.