Derivatives and Graphing

Increasing and Decreasing Where a function is increasing...

Increasing and Decreasing …tangent lines have positive slope.

Increasing and Decreasing So the first derivative is positive.

Increasing and Decreasing Where the function is decreasing...

Increasing and Decreasing …tangent lines have negative slope.

Increasing and Decreasing So the first derivative is negative.

Increasing and Decreasing Where a function changes from increasing to decreasing…

Increasing and Decreasing …or from decreasing to increasing…

Increasing and Decreasing A function may have local maximum or minimum values.

Increasing and Decreasing If the first derivative exists at these points, it is zero.

Increasing and Decreasing Local max Local min Local extrema may also exist where the derivative is undefined.

Identify the open intervals on which the function is increasing or decreasing.

Identify the open intervals on which the function is increasing or decreasing.

Find the critical numbers, find the open intervals on which the function is increasing or decreasing and locate all relative extrema

Find the critical numbers, find the open intervals on which the function is increasing or decreasing and locate all relative extrema

Concavity inflection point inflection point concave down concave down concave up

Concavity Where the curve is concave down... ...slopes of tangent lines are decreasing.

Concavity Since the first derivative is decreasing... concave down …the second derivative is negative.

Concavity Where the curve is concave up... concave down concave up ...slopes of tangent lines are increasing.

Concavity Since the first derivative is increasing... concave down concave up …the second derivative is positive.

Concavity concave down At the inflection points, the second derivative (if it exists) is 0. concave down concave up

Concavity concave down concave down concave up

Summary

Summary

Identify the open intervals on which the function is concave upward and the intervals where the function is concave down.

Identify the open intervals on which the function is concave upward and the intervals where the function is concave down.

Find all relative extrema.

Find all relative extrema.