6HGraphs

= Graphs of the Antiderivatives of Functions =


 * Recall that the words ** integral ** and ** antiderivative ** refer to the same process


 * and that they are the __ reverse __ of ** derivative **.


 * So if we let F(x) be the ** antiderivative ** of f(x)

math . \qquad F(x)= \displaystyle{ \int f(x) \; dx} \qquad. math


 * then it is also true that f(x) is the ** derivative ** of F(x)

math . \qquad f(x) = \dfrac{d}{dx} \Big( F(x) \Big) \qquad. math


 * ** BUT ** the process of taking the derivative loses a piece of information -- the value of the constant term.
 * ** {Remember, this is why we have to add c when finding indefinite integrals} **


 * This means that, when it comes to graphs:
 * If we start with the graph of a function,
 * we can __ accurately __ sketch the graph of its ** derivative **.


 * BUT


 * If we start with the graph of a function, f(x),
 * we can accurately sketch the __ shape __ of its ** antiderivative **, F(x) +c,
 * but we can't know where that shape should be placed vertically on the cartesian axes
 * because we don't know what the constant was.


 * Therefore the antiderivative, F(x) + c represents a ** f amily of curves **
 * all with the same shape but translated up or down by different amounts.
 * ie the family of curves is an infinite set.
 * There are an infinite number of possible curves F(x) + c, due to an infinite number of possible values for c.


 * We tend to place the shape of F(x) somewhere on the axes and call it ** __an__ ** antiderivative of f(x).


 * One point (one set of coordinates) that F(x) passes through is enough to determine the value for c
 * and hence where ** __the__ ** antiderivative should be drawn.


 * Features of the Graphs of Antiderivatives **

math . \qquad \text{For } \, F(x)= \displaystyle{ \int f(x) \; dx} \qquad. math

... ...


 * Example **

... ... Consider the following graph of a function, f(x). ... ... Key features of the graph have been annotated. ... ...

... ... This information can now be used to sketch an antiderivative graph, F(x), (in red).

... ....

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