Section 4.8

Optimization

We don't just want *an* answer—we want the *best* answer. Maximizing profit, minimizing waste, optimizing trajectories. This is where calculus pays the bills.

1

The 3-Step Strategy

  1. 1
    Primary Equation

    What are you maximizing or minimizing? Write a formula for it (e.g., ).

  2. 2
    Constraint Equation

    Use the limitation given (e.g., 2400ft of fence) to eliminate variables until your primary equation has only one variable.

  3. 3
    Calculus

    Find the derivative, set it to zero, and verify using the First or Second Derivative Test.

2

Worked Example: The Farmer

Maximizing Area

A farmer has 2400 ft of fencing to enclose a rectangular area bordering a straight river. No fence is needed along the river. Find dimensions that maximize area.

Dynamic Fence

River View
A = 400,000
Step 1 & 2: Equations
  • Constraint:
  • Primary:
Step 3: Derivatives
|
Solution

Width ft.
Length ft.
Max Area: .

3

Level Up Examples

Example A: The Open Box

A box is made by cutting squares of size from corners of a sheet. Maximize volume.

1. Formula:

2. Optimize:

.
3. Check Logic:

means we cut entire side (V=0). Must be min.

Max at .

Example B: Efficient Numbers

Find two positive numbers whose product is 100 and whose sum is minimal.

1. Constraint:
2. Primary (Sum):
3. Optimize:

Answer: 10 and 10.

5

Practice Quiz

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