Calculus / Unit 6: Applications of Derivatives II
6.5 Optimization
Notes
Optimization means finding the largest or smallest value of a quantity.
- Steps: draw a picture, write the quantity to optimize, use a constraint to write it with one variable, find its domain, find critical numbers, and check the endpoints or the second derivative.
- Example: two numbers add to 20. Product $P = x(20 - x)$, $P' = 20 - 2x = 0$ at $x = 10$. Max product is 100.
Practice
Two numbers add to 20. What is the largest possible product?
$P = x(20 - x)$ is maximized at $x = 10$, giving 100.
A rectangle has perimeter 100 ft. What is its maximum area?
$A = x(50 - x)$ is maximized at $x = 25$ (a square), giving 625.
A farmer has 120 ft of fence for three sides of a rectangle (a river forms the fourth side). What is the maximum area?
With width $x$, length is $120 - 2x$, so $A = 120x - 2x^2$, $A' = 120 - 4x = 0$ at $x = 30$. The length is 60 and the area is $30 \cdot 60 = 1800$.
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