Answer:
[C] 0
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Distributive Property
Algebra I
- Terms/Coefficients
- Functions
- Function Notation
Calculus
Limits
Derivatives
Definition of a Derivative: [tex]\displaystyle f'(x)= \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle\displaystyle g(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[/tex]
[tex]\displaystyle f(x) = \frac{3}{5x^4 + 3}[/tex]
[tex]\displaystyle g(0)[/tex]
Step 2: Differentiate
- Substitute in x [Function g(x)]: [tex]\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{h}[/tex]
- Substitute in function f(x) [Function g(x)]: [tex]\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{\frac{3}{5(0 + h)^4 + 3} - \frac{3}{5(0)^4 + 3}}{h}[/tex]
- Simplify: [tex]\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{\frac{3}{5h^4 + 3} - 1}{h}[/tex]
- Rewrite: [tex]\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{3}{h(5h^4 + 3)} - \frac{1}{h}[/tex]
- Rewrite: [tex]\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{3}{h(5h^4 + 3)} - \frac{5h^4 + 3}{h(5h^4 + 3)}[/tex]
- [Subtraction] Combine like terms: [tex]\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{3 - 5h^4 + 3}{h(5h^4 + 3)}[/tex]
- [Addition] Simplify: [tex]\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{-5h^4 + 6}{h(5h^4 + 3)}[/tex]
- [Distributive Property] Distribute h: [tex]\displaystyle\displaystyle g(0) = \lim_{h \to 0} \frac{-5h^4 + 6}{5h^5 + 3h}[/tex]
- Evaluate limit [Power Method]: [tex]\displaystyle\displaystyle g(0) = 0[/tex]
Since the bottom polynomial has a higher degree than the top polynomial, the bottom polynomial will increase faster.
∴ If the bottom is approaching a bigger value, the fraction gets smaller and smaller, approaching 0.
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
