Answer:
See explanation.
General Formulas and Concepts:
<u>Algebra I</u>
- Terms/Coefficients
- Factoring
<u>Algebra II</u>
<u>Pre-Calculus</u>
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]:
Derivative Property [Addition/Subtraction]:
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Quotient Rule]:
Parametric Differentiation:
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
<u>Step 2: Find Derivative</u>
- [<em>x</em>] Differentiate [Basic Power Rule and Quotient Rule]:
- [<em>y</em>] Differentiate [Basic Power Rule and Quotient Rule]:
- Substitute in variables [Parametric Derivative]:
- [Parametric Derivative] Simplify:
- [Parametric Derivative] Polynomial Long Division:
- [Parametric Derivative] Factor:
Here we see that if we increase our values for <em>t</em>, our derivative would get closer and closer to 0.5 but never actually reaching it. Another way to approach it is to take the limit of the derivative as t approaches to infinity. Hence .
Topic: AP Calculus BC (Calculus I + II)
Unit: Parametrics
Book: College Calculus 10e