07 5curves

Graphs of the Antiderivatives of Functions

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

  • and they are the reverse of derivative.

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

… … $F(x)= \displaystyle{ \int f(x) \; dx}$
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then it is also true that f(x) is the derivative of F(x)

… … $f(x) = \dfrac{d}{dx} \Big( F(x) \Big)$

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BUT the process of taking the derivative loses a piece of information:

  • the value of the constant term is lost when we take the derivative.
  • This is why we have to add +c when finding indefinite integrals.

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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, $f \big( x \big)$,
    • we can accurately sketch the shape of its antiderivative, $F \big( x \big) +c$,
    • but we can't know where that shape should be placed vertically on the Cartesian axes
      • because we don't know the value of the constant (+c).

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Therefore the antiderivative, $F \big( x \big) + c$ represents a family of curves

  • all with the same shape but translated up or down by different amounts.
  • There are an infinite number of possible curves $F\big( x \big) + c$,
    • due to an infinite number of possible values for c.
  • ie the family of curves is an infinite set.

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When sketching $F \big( x \big)$, we place the shape of $F \big( x \big)$ somewhere on the axes and call it an antiderivative of $f \big( x \big)$.

  • I usually try to place it so the antiderivative graph doesn't overlap the graph of the original function (or keep the overlap to a minimum)
  • Some people draw it with a y-intercept of $(0, \; 0)$

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

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Features of the Graphs of Antiderivatives

… For $F \big( x \big) = \displaystyle{ \int } f\big( x \big) \; dx$

07.5table1.GIF

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Example

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Consider the following graph of a function, $f \big( x \big)$.

  • Key features of the graph have been annotated.
07.5graph1.gif

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

07.5graph2.gif

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