Lesson 11: 5.11 Solving Optimization Problems
Duration of Days: 2
Lesson Objective
By the end of this section, students will be able to:
Construct Complex Models: Link coordinate geometry (points on a curve) to geometric measurements (area/volume).
Simplify Objectives: Recognize when a function can be simplified before differentiating (e.g., ignoring the square root in distance problems).
Define Feasible Domains: Mathematically state the interval for which the variable makes sense.
Verify Absolutes: Provide a written justification that proves the local extremum found is the global extremum for the problem's context.
How do we optimize shapes inscribed within other shapes (e.g., a rectangle inside a parabola)?
When the "cost" of different sides of a shape varies, how does that shift the optimal dimensions?
How can we use the Distance Formula as an objective function to find the closest point on a curve?
Does the "absolute" maximum always occur at a critical point, or must we always check the endpoints of the feasible domain?
nscribed Figure
Feasible Domain (Physical Constraints)
Objective Function (Simplified)
Distance Formula
Cost Function
Candidates Test
Global (Absolute) Extremum
Surface Area vs. Volume
Derivative of the Square
FUN-4.B: Calculate minimum and maximum values in applied contexts.
FUN-4.C: Interpret minimum and maximum values in applied contexts.
Interpretive Goal: Determine the reasonableness of solutions within the context of the problem.
In this second half of optimization, the functions become non-linear. Students will tackle:
Inscribed Figures:
Finding the largest area of a rectangle that can fit under a curve Distance Problems.
Cost/Material Optimization: Designing containers where the base material costs more than the side material.
Rigorous Justification: Using the Candidates Test (checking endpoints and critical points) or the Second Derivative Test for Absolute Extrema to prove their answer is the absolute best.
Purpose: To develop the persistence required for multi-step AP Calculus problems. These problems are often the "long-form" questions on the exam that separate a score of 4 from a 5.
DOK Level 3 (Strategic Thinking): Selecting the correct variables and formulas to represent a physical constraint.
DOK Level 4 (Extended Thinking): Generalizing a solution (e.g., finding the optimal ratio of height to radius for any cylinder of a fixed volume).
For Struggling Learners (Scaffolding):
Geometric Formula Bank: Provide a reference sheet for Volume/Surface Area of cylinders, cones, and spheres so they don't get "stuck" on the geometry.
"Step 0": Have students draw the scenario on a coordinate plane before writing any equations.
Partial Key: Provide the "Objective Function" for the most difficult problems, allowing them to focus on the Calculus (differentiation and solving).
For Advanced Learners (Extension): Introduce a basic "Snell’s Law" optimization problem (finding the fastest path between two points in different media).
Surface Area Challenges: Optimize a can design that includes a "seam" allowance (adding a constant to the surface area formula).
AP College Board Assessments