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
Related rates problems expand the applications of implicit differentiation. Many (not all!) related rates problems present a quantity changing with respect to time, usually denoted as the variable t. Use of the Chain Rule (whether or not time is the independent variable) becomes important for the correct resolution of these problems. We are spending multiple days on Topics 4.4 and 4.5 to expose students not only to the wide variety of possible contexts, but also to illustrate the similar structure of a complete solution.
Because students have recently been practicing implicit differentiation, the concept of including a dV/dt term (or dz/dt or dh/dt term) in their derivative equations should not cause confusion. The habit of writing equations pays off big in this lesson as students may need to substitute numeric values on both sides of the derivative equation. That’s hard to do without writing an equation in the first place! We believe a carefully labeled diagram of the original problem will aid in writing initial equations to relate the variables. (The only numeric values in your diagram should be those that are constant throughout the entire problem --- values that vary throughout the problem should be noted with variables in the diagram.)
Related rates problems on the AP Calculus Tests may appear with or without an attached context! If the “related rate” is embedded in a story problem, units become an important detail to include.
See the following examples for extra practice in your classroom.
2019 AB 4
2002 AB 5
2002 AB 6 (Form B)
1995 AB 5
1985 AB 9 (multiple choice)
1998 AB 40 (multiple choice)
Students must be careful to substitute numeric values only after they have completed writing their derivative equation. Premature substitution can lead to incorrect solutions as terms may simply vaporize to zero when taking the derivative of the constant term.
Those who have forgotten the formulas for area or volume or trig relationships will have a little extra work to do within related rates problems. Cylinders, cones, and tangents abound in these exercises! Our exercises incorporate spheres, cylinders and the Pythagorean Theorem.