Answer:
[C] 27
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Algebra I
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 \lim_{h \to 0} \frac{f(3 + h) - f(3)}{h}[/tex]
[tex]\displaystyle f(x) = x^3[/tex]
Step 2: Solve
- Substitute in function value: [tex]\displaystyle \lim_{h \to 0} \frac{(3 + h)^3 - 3^3}{h}[/tex]
- Evaluate exponents: [tex]\displaystyle \lim_{h \to 0} \frac{(3 + h)^3 - 27}{h}[/tex]
- Expand: [tex]\displaystyle \lim_{h \to 0} \frac{h^3 + 9h^2 + 27h + 27 - 27}{h}[/tex]
- [Subtraction] Combine like terms: [tex]\displaystyle \lim_{h \to 0} \frac{h^3 + 9h^2 + 27h}{h}[/tex]
- Factor: [tex]\displaystyle \lim_{h \to 0} \frac{h(h^2 + 9h + 27)}{h}[/tex]
- Simplify: [tex]\displaystyle \lim_{h \to 0} h^2 + 9h + 27[/tex]
- Evaluate limit: [tex]\displaystyle 0^2 + 9(0) + 27[/tex]
- Evaluate exponents: [tex]\displaystyle 9(0) + 27[/tex]
- Multiply: [tex]\displaystyle 27[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
