The vertices of a polygon are A(0,4), B(5,4), C(5,-3) and D(0,-3). Graph the vertices and connect them in order. What is the per
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Answer:
The company should spend $40 to yield a maximum profit.
The point of diminishing returns is (40, 3600).
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
- Left to Right
Equality Properties
- Multiplication Property of Equality
- Division Property of Equality
- Addition Property of Equality
- Subtraction Property of Equality
<u>Algebra I</u>
Coordinate Planes
- Coordinates (x, y) → (s, P)
Functions
- Function Notation
Terms/Coefficients
- Factoring/Expanding
Quadratics
<u>Algebra II</u>
Coordinate Planes
- Maximums/Minimums
<u>Calculus</u>
Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]:
Derivative Property [Addition/Subtraction]:
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
1st Derivative Test - tells us where on the function f(x) does it have a relative maximum or minimum
- Critical Numbers
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
<u>Step 2: Differentiate</u>
- [Function] Derivative Property [Addition/Subtraction]:
- [Derivative] Rewrite [Derivative Property - Multiplied Constant]:
- [Derivative] Basic Power Rule:
- [Derivative] Simplify:
<u>Step 3: 1st Derivative Test</u>
- [Derivative] Set up:
- [Derivative] Factor:
- [Multiplication Property of Equality] Isolate <em>s </em>terms:
- [Solve] Find quadratic roots:
∴ <em>s</em> = 0, 40 are our critical numbers.
<u>Step 4: Find Profit</u>
- [Function] Substitute in <em>s</em> = 0:
- [Order of Operations] Evaluate:
- [Function] Substitute in <em>s</em> = 40:
- [Order of Operations] Evaluate:
We see that we will have a bigger profit when we spend <em>s</em> = $40.
∴ The maximum profit is $3600.
∴ The point of diminishing returns is ($40, $3600).
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation (Applications)