07Agradient

toc = Sketching the Gradient Function from a Graph =

Given the graph of a function y = f(x) we can sketch the gradient function denoted by f ' (x).

The gradient function of a polynomial is one degree less than the original function.

To sketch a gradient function consider each section of the original graph

Conditions of differentiability
Recall (from Conditions of Differentiability) that the gradient function or derivative, f ' (x) exists for a given value of x only if the graph of f(x) is __**smooth**__ and __**continuous**__ at that x-value. It must be possible to draw a unique tangent at that x-value.

This means that the gradient function (derivative) does not exist (is not defined) in the following conditions:
 * where there is a vertical asymptote
 * where there is a break or a hole
 * where there is a sharp corner (cusp)
 * at the endpoints of the domain

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Gradient Functions Worksheet
For each of the functions shown, sketch the graph of its gradient function. State the domain of each gradient function.

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