Unit 3

Curve Sketching

Lesson 4: Concavity and Points of Inflection

Watch the video Examples of Curve Sketching for additional examples. Take time to work through these examples to have a better understanding of the information obtained from the first and second derivatives.

While working through each example, use the following steps as a guide in when sketching the curve.

Step 1:
Find the critical points and plot them. This is done by setting the first derivative equal to zero, solving for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math», and then finding the corresponding «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-coordinates.

Step 2:
Test the regions on either side of the critical points to identify the intervals increase of decrease. Use an interval chart or the sign analysis method.

Step 3:
Find the possible inflection points and plot them. This is done by setting the second derivative equal to zero, solving for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math», and then finding the corresponding «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-coordinates. These points are considered possible inflection points until concavity has been determined.

Step 4:
Test the regions on either side of the possible inflection points to identify concavity. Use an interval chart or the sign analysis method. Where there is a change in concavity, there is an inflection point.

Step 5:
Sketch the curve using the information obtained in Steps 1 to 4. It may be helpful to find other points such as the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercepts.