Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
Derivatives and Shapes of Graphs- DA4
Objectives
- Explain how the sign of the first derivative affects the shape of a function’s graph.
- State the first derivative test for critical points.
- Use concavity and inflection points to explain how the sign of the second derivative affects the shape of a function’s graph.
- Explain the concavity test for a function over an open interval.
- Explain the relationship between a function and its first and second derivatives.
- State the second derivative test for local extrema.
Summary
The critical numbers of a continuous function are the values of for which or does not exist. These values are important because they identify horizontal tangent lines or corner points on the graph, which are the only possible locations at which a local maximum or local minimum can occur.
Given a differentiable function whenever is positive, is increasing; whenever is negative, is decreasing. The first derivative test tells us that at any point where changes from increasing to decreasing, has a local maximum, while conversely at any point where changes from decreasing to increasing has a local minimum.
Given a twice differentiable function if we have a horizontal tangent line at and is nonzero, the sign of tells us the concavity of and hence whether has a maximum or minimum at In particular, if and then is concave down at and has a local maximum there, while if and then has a local minimum at If and then the second derivative does not tell us whether has a local extreme at or not.