Sketching the Gradient Function from a Graph

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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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• For each of the functions shown, sketch the graph of its gradient function. State the domain of each gradient function.

1) Solution .

2) Solution .

3) Solution .

4) Solution .

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