The graph below shows \(f'(x)\) – the rate of change of \(f(x)\). Then we would look at the values of \(F\) at the endpoints to find which was the global min. It's the only critical point, so it must be a global max. \(f = F'\) goes from positive to negative there, so \(F\) has a local max at that point. Note that this is a different way to look at a problem we already knew how to solve – in Chapter 2, we would have found critical points of \(F\), where \(f = 0\): there's only one, when \(t = 3\). ![]() The maximum value is when \(t = 3\) the minimum value is when \(t = 0\). The area between \(t = 3\) and \(t = 4\) is much smaller than the positive area that accumulates between 0 and 3, so we know that \(F(4)\) must be larger than \(F(0)\). Since \( F(b) = F(a) \int\limits_a^b F'(x) \,dx \), we know that \(F\) is increasing as long as the area accumulating under \(F' = f\) is positive (until \(t = 3\)), and then decreases when the curve dips below the \(x\)-axis so that negative area starts accumulating. Where does \(F(t)\) have maximum and minimum values on the interval ? There are no small families in the world of antiderivatives: if \(f\) has one antiderivative \(F\), then \(f\) has an infinite number of antiderivatives and every one of them has the form \(F(x) C\). This process is called antidifferentiation or integration. ![]() ![]() We antidifferentiate, or integrate, or find the indefinite integral of a function. The function \(f\) is still called the integrand. The \( \int \) symbol is still called an integral sign the \(dx\) on the end still must be included you can still think of \( \int \) and \(dx\) as left and right parentheses. But in this notation, there are no limits of integration. This notation resembles the definite integral, because the Fundamental Theorem of Calculus says antiderivatives and definite integrals are intimately related. The antiderivative is also called the indefinite integral. The antiderivative of a function \(f(x)\) is a whole family of functions, written \(F(x) C\), where \(F'(x) = f(x)\) and \(C\) represents any constant.
0 Comments
Leave a Reply. |