Calculate and interpret related rates in applied contexts.
I can identify constants and variables in an applied context.
I can apply the chain rule when differentiating multiple variables in a related rate problem
Quick Lesson Plan
We continue our study of related rates in this lesson by focusing on right circular cones that are being filled and drained. The proportional relationship between radius and height will provide the needed substitutions for solving related rates problems today. The independent variable continues to be time, t, and our derivatives will be computed with respect to this variable.
Continue to reinforce the need to apply the chain rule so that a complete derivative equation can be written. Students will be challenged indeed to solve for dr/dt or dh/dt if that term doesn’t appear in their derivative expressions! Work intentionally through the debrief portion of today’s lesson. There will still remain ample time to both emphasize the Important Ideas and have students check their understanding with problems 1 and 2. We are assigning FRQs to expose students to the rigor and complexity of potential related rates questions on their AP Calculus Test.
See Day 7 notes for notes and suggested FRQs.
For some students, this is the lesson that finally convinces them of the importance of the chain rule. However, many types of mathematics come together in a related rates problem: geometry, trigonometry, algebra, and calculus. Remembering formulas and relationships from previous classes will be a benefit, as will be strong solving skills, numerical accuracy, and the ability to interpret a solution in context.