Slope at Point of Tangency

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Presentation transcript:

Slope at Point of Tangency Mr. Miehl www.stoga.net miehlm@tesd.net

Objective To determine the slope of a function at a given point. To determine the slope of a tangent line to a function at a given point. Both are done the same way!

The Derivative is… Computed by finding the limit of the difference quotient as ∆x approaches 0. Used to find the slope of a function at a point. Used to find the slope of the tangent line to a graph f (x), and is usually denoted f’(x). Used to find the instantaneous rate of change of a function.

Limit Definition of the Derivative Use the limit definition to find the derivative of:

Limit Definition of the Derivative CAUTION: Possible mistakes ahead!

Limit Definition of the Derivative

Limit Definition of the Derivative

Limit Definition of the Derivative A formula for finding the slope of the tangent line of f (x) at a given point.

Slope of a Function Find the slope of at .

Slope of a Function

Slope of a Function

Slope of a Function Find the slope of at .

Slope of Tangent Line Find the slope of the tangent line to the graph of at (–2, 16).

Slope of Tangent Line

Slope of Tangent Line Find the slope of the tangent line to the graph of at (–2, 16).

Value of the Derivative Find the value of the derivative of

Value of the Derivative

Value of the Derivative

Value of the Derivative

Value of the Derivative

Value of the Derivative Find the value of the derivative of

Value of the Derivative Find if .

Value of the Derivative

Value of the Derivative

Value of the Derivative Find if .

Conclusion The derivative is a formula used to find the slope of function or slope of the tangent line to a function. To find the slope of a function or slope of the tangent line to a function, first, find the derivative and, second, plug the corresponding x-value into the derivative.