Preview Activity 5.2.1.
Consider the function \(A\) defined by the rule
\begin{equation*}
A(x) = \int_1^x f(t) \, dt\text{,}
\end{equation*}
where \(f(t) = 4-2t\text{.}\)
(a)
(b)
Use the First Fundamental Theorem of Calculus to find a formula for \(A(x)\) that does not involve integrals. That is, use the first FTC to evaluate \(\int_1^x (4-2t) \, dt\text{.}\)
(c)
(d)
Using the formula you found in (b) that does not involve integrals, compute \(A'(x)\text{.}\)
(e)
While we have defined \(f\) by the rule \(f(t) = 4-2t\text{,}\) it is equivalent to say that \(f\) is given by the rule \(f(x) = 4 - 2x\text{.}\) What do you observe about the relationship between \(A\) and \(f\text{?}\)
