Let F(x) = ∫_{0}^{x} t^2 dt. What is F'(x)?

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Multiple Choice

Let F(x) = ∫_{0}^{x} t^2 dt. What is F'(x)?

Explanation:
Think of the Fundamental Theorem of Calculus: if F(x) is the integral from a constant to x of f(t) dt, then F'(x) = f(x). Here the integrand is t^2, so F'(x) = x^2. You can also confirm by computing F(x) = ∫_0^x t^2 dt = x^3/3, whose derivative is x^2.

Think of the Fundamental Theorem of Calculus: if F(x) is the integral from a constant to x of f(t) dt, then F'(x) = f(x). Here the integrand is t^2, so F'(x) = x^2. You can also confirm by computing F(x) = ∫_0^x t^2 dt = x^3/3, whose derivative is x^2.

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