Answer :
The specific antiderivative F of the function f(t) = [tex]t^2 + 2t[/tex], where F(6) = 84, is F(t) = [tex](1/3)t^3 + t^2 + C[/tex], where C is a constant.
To find the antiderivative F of the function f(t), we need to integrate f(t) with respect to t. The integral of [tex]t^2\ is\ (1/3)t^3[/tex], and the integral of 2t is [tex]t^2[/tex]. Adding these two results, we obtain the antiderivative F(t) = [tex](1/3)t^3 + t^2[/tex]. However, since an antiderivative represents a family of functions, we need to add a constant of integration, denoted as C.
Given that F(6) = 84, we can substitute t = 6 into the expression for F(t) and solve for C. Plugging in 6, we have 84 = [tex](1/3)(6)^3 + (6)^2 + C[/tex]. Simplifying this equation, we get 84 = 72 + 36 + C. Solving for C, we find that C = -24.
Thus, the specific antiderivative F(t) of f(t) = t^2 + 2t, with F(6) = 84, is F(t) = [tex](1/3)t^3 + t^2 - 24[/tex].
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