How to find f'(x) graphs from f(x) graphs and vice versa (1 Viewer)

fan96

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With these sorts of questions it may help to remember your definitions and think things through slowly and methodically.

is the gradient function. That means that the value of is the gradient of at .

A gradient of 0 means that there is a stationary point.

As the gradient approaches zero, the graph approaches a horizontal line. If it approaches zero from the positive side, then you know that has a maximum there, because the gradient is always positive (i.e. the graph is going up) before it reaches zero and vice versa for the negative side.

A positive gradient means the graph is going up.

A negative gradient means the graph is going down.

As the gradient approaches infinity, the graph approaches a vertical line (whether it goes up or down depends on if its positive or negative infinity). This means a local maximum/minimum in the graph of means the graph of is (relatively) steepest at that point, because as approaches its (local) maximum/minimum value, the graph of gets increasingly steeper.

A good way to visualise this is to look at graphs of a polynomial and its derivatives, and examine what happens to the polynomial when its derivative is approaching zero or approaching a local minimum/maximum.
 
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