Question

• Let f be a function such that f(-2) = 8 and f'(-2) = 4.

Answer 1

Answer:

-2

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Calculus

Derivatives

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Quotient Rule: $\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

$\frac{d}{dx} [\frac{f(x)}{h(x)} ] \ at \ x = -2\\h(x) = x^3\\f(-2) = 8\\f'(-2) = 4$

Step 2: Differentiate

1. Differentiate [Quotient Rule]:                    $\frac{d}{dx} [\frac{f(x)}{h(x)} ] = \frac{f'(x)h(x) - f(x)h'(x)}{h(x)^2}$
2. Differentiate h(x) [Basic Power]:                h'(x) = 3x²

Step 3: Evaluate

1. Define differential:                     $\frac{f'(x)x^3 + f(x)[3x^2]}{(x^3)^2}$
2. Substitute in variables:             $\frac{f'(-2)h(-2) + f(-2)h'(-2)}{h(-2)^2}$
3. Substitute in variables:             $\frac{4(-2)^3 - 8[3(-2)^2]}{[(-2)^3]^2}$
4. Evaluate:                                    -2