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2.2 Graphs of Antiderivative Functions

Deducing Antiderivative Function Graphs

Method 1

i) The sign of f^{\prime}(x) ( >0 or <0) decides the sloping (upward or downward) of f(x).

ii) Any x-intercept of f^{\prime}(x) (i.e. f^{\prime}(x)=0) shows the stationary point of f(x). If f^{\prime}(x) changes sign before and after passing the x-axis, then it is a turning point for f(x). If not, it is just a stationary point.

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2.1 Graphs of Derivative Functions

Deducing Derivative Function Graphs

  • Here, it is assumed that the function of the original function is unknown, only its graph.
  • Given the graph of a function, some facts can be told about the derivative, recalling the fact that the derivative function represents the instantaneous rate of change of the original function at different points.
  • Firstly, it is obvious that the original function and its derivative would usually have the same domain, except when certain x values (or range) where the function is not differentiable.
  • It is also logical that the graph of derivative follows that of the original. For instance, if the original function is a polynomial, so will the derivative, thus its graph would look like one.
  • We can then look at specific points. If there is(are) stationary point(s) found, we know that f^{\prime}(x)=0 at these point(s). This is because the graph f(x) at these points are somewhat horizontal, thus it’s instantaneous rate of change is 0.
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