Use derivatives to solve optimization problems.
Interpret maximums and minimums in applied contexts.
I can calculate maximum and minimum values in applied contexts.
I can interpret maximum and minimum values in applied contexts.
Quick Lesson Plan
In this lesson, students work to find a La Croix can that minimizes cost. First, students explore multiple possible dimensions that satisfy a constraint. Then they represent these graphically and relate the lowest possible cost to the minimum of the function. Next, students use calculus to find this minimum cost by finding critical values and applying the first or second derivative test. Optimization problems provide interesting and challenging real-world application of the calculus students have been studying in the classroom.
We recommend having students work through the first page only, then debriefing the first page before having students move to the analytical portion of the lesson on page 2. Here are some ideas you’ll want to emphasize when debriefing the front page:
· Volume is staying constant in all three cans.
· Adjusting the radius requires an adjustment to the height as well. Since the volume is staying constant, we can always solve for the height by dividing volume by (pi*r^2).
· Ask several students to explain what a particular dot represents on this curve. We want students to understand that each dot represents the cost of producing a can at a specific radius. This will help students make the transition to writing an equation C(r) on the next page that gives the cost as a function of the radius.
· Ask students to estimate what they think the lowest cost is. Ask them if it’s possible that one of their dots on the graph already gives the lowest cost. Ask them if it’s possible that there is a cost even lower than the ones shown on the graph by their dots. (You might want to add some additional dots to the graph for r<1 since this makes the “Nike swoosh” feature of the graph. It is likely that students will not have chosen this value for r on their own.)
Model a consistent approach to solving optimization problems and remind students often why an organized, systematic method must be followed: while context can vary widely from one problem to the next, the strategy for solving the problems should not. Use consistently the steps outlined in the Important Ideas section to demonstrate a straightforward approach to these problems.
Optimization problems are presented in many formats on the AP Test. Students may see graphical, analytical, and numerical multiple choice questions along with lengthier context or story problems as an FRQ. We are guaranteed, however, that students will be required to identify and justify the location and characteristics of function extrema in a variety of contexts on the AP Test. Using the first and second derivative tests, as well as the candidates test, is a capstone in their study of differential calculus.
As with our work with related rates, the geometry, algebra, and trigonometry required to construct an equation in one variable can be the biggest challenge for students. Once they have written an equation for the quantity to be maximized or minimized, the analytical work often flows smoothly. Writing a precise conclusion to their work may require discussion of more than one variable (radius and height, for example), so students need to be attentive to the original question. “A story problem needs a story answer,” is the mantra for writing complete solutions.