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  2. AP Calculus
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How do you evaluate ∫abf(x)dx\int_{a}^{b} f(x) dx∫ab​f(x)dx using FTOC Part 2?

  1. Find the antiderivative F(x) of f(x). 2. Evaluate F(b) and F(a). 3. Subtract F(a) from F(b): F(b) - F(a).
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How do you evaluate ∫abf(x)dx\int_{a}^{b} f(x) dx∫ab​f(x)dx using FTOC Part 2?

  1. Find the antiderivative F(x) of f(x). 2. Evaluate F(b) and F(a). 3. Subtract F(a) from F(b): F(b) - F(a).

How do you find g′(x)g'(x)g′(x) if g(x)=∫axf(t)dtg(x) = \int_{a}^{x} f(t) dtg(x)=∫ax​f(t)dt?

Apply FTOC Part 1: g′(x)=f(x)g'(x) = f(x)g′(x)=f(x).

How do you handle a constant of integration when using FTOC Part 2?

The constant of integration cancels out when evaluating F(b) - F(a), so it's not necessary to include it.

How to find g′(x)g'(x)g′(x) if g(x)=∫ah(x)f(t)dtg(x) = \int_{a}^{h(x)} f(t) dtg(x)=∫ah(x)​f(t)dt?

Apply FTOC Part 1 and the chain rule: g′(x)=f(h(x))∗h′(x)g'(x) = f(h(x)) * h'(x)g′(x)=f(h(x))∗h′(x)

Define definite integral.

An integral with upper and lower limits, resulting in a numerical value.

What is an antiderivative?

A function whose derivative is the original function.

Define the Fundamental Theorem of Calculus.

A theorem that connects the derivative and the integral, stating that differentiation and integration are inverse processes.

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.