exercise:05c9c52e1f

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Let [math]F(t) = \int_0^t (6x^2 - 4x + 1) \; dx[/math].

• Using just the Fundamental Theorem and without evaluating [math]F[/math], find [math]F^\prime(t)[/math], [math]F^\prime(-1)[/math], [math]F^\prime(2)[/math], and [math]F^\prime(x)[/math].
• Find [math]F(t)[/math] as a polynomial in [math]t[/math] by finding a polynomial which is an antiderivative of [math]6x^2 - 4x + 1[/math].
• Differentiate the answer in (b) and thereby check (a).