L4 Concavity and Points of Inflection - Part 2
Completion requirements
Unit 3
Curve Sketching
Lesson 4: Concavity and Points of Inflection
The first derivative is fundamental in determining where a function increases and decreases. In previous lessons, techniques for using the first derivative as a tool in curve sketching were introduced. As such, you should now be able to
Curvature is another important feature of the graph of a function. Calculus provides tools for analyzing the concavity of a curve. The sign of the second derivative tells whether a curve opens downward or upward, and thus aids in sketching more accurate graphs of the function.
- identify, from a sketch, locations on the graph of a function where the first derivative is zero or undefined,
- relate the zeros of the derivative function to the critical points of the original function,
- explain the circumstances that lead to a local maximum or minimum where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8800;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math»,
- explain why «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»a«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» does not automatically guarantee the existence of a maximum or minimum at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mi»a«/mi»«/mrow»«/mstyle»«/math»,
- verify whether a point is a maximum or minimum,
- use the first derivative to locate local extrema to aid in the sketching of graphs,
- explain the difference between local and absolute extrema, and
- explain how the sign of the first derivative indicates whether a function is increasing or decreasing.
Curvature is another important feature of the graph of a function. Calculus provides tools for analyzing the concavity of a curve. The sign of the second derivative tells whether a curve opens downward or upward, and thus aids in sketching more accurate graphs of the function.