Overview
This lecture covers optimization problems in calculus, focusing on maximizing or minimizing functions in contexts such as productivity, drug dosage, geometry, and profit.
Maximizing Productivity
- Productivity as a function of years of service is optimized by finding critical values of its derivative.
- The first derivative is set to zero: -8T + 224 = 0; solving gives T = 28.
- Second derivative is negative, indicating a maximum at T = 28.
- Productivity is greatest when T = 28 years, yielding maximum output.
Drug Dosage and Maximum Blood Pressure
- Blood pressure function maximized with respect to drug dosage.
- First derivative set to zero; factored to find critical values at X = 0 and X = 1/9.
- First derivative test confirms X = 1/9 is a relative maximum.
- Maximum blood pressure occurs when dosage is X = 1/9 (≈0.111 cc).
Maximize Function with Constraints
- Maximize b = xy² with constraint x + y² = 9, x > 0, y > 0.
- Substitute x = 9 - y² to get b in one variable.
- Take first and second derivatives; solve for critical values.
- Valid maximum at y = 3/√2, x = 9 - (3/√2)² = 4.5.
- Maximum value of b is 81/4.
Maximizing Rectangular Area (Swim Area Problem)
- Area A = xy, constrained by perimeter 2x + y = 1800 (since one side is on the shoreline).
- Substitute y = 1800 - 2x, so A = x(1800 - 2x).
- Derivative set to zero gives x = 450, y = 900.
- Maximum area is 450 × 900 = 405,000 square yards.
Maximum Profit Problem
- Profit P = Revenue - Cost; profit function is P(x) = 35x - 0.1x² - 20.
- Set derivative P'(x) = -0.2x + 35 to zero; solve for x = 175.
- Second derivative is negative; confirms maximum at x = 175 units.
- Maximum profit when producing/selling 175 units is $3,042.50.
Key Terms & Definitions
- Critical Value — Where the first derivative is zero or undefined; used to find maxima/minima.
- Objective Function — The function to be maximized or minimized.
- Constraint — An equation or inequality that limits variables in optimization.
- Second Derivative Test — Determines concavity to confirm maxima or minima.
Action Items / Next Steps
- Practice setting up and solving optimization problems with constraints.
- Review how to apply the first and second derivative tests for maxima and minima.
- Complete assigned homework problems on optimization using calculus.