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galben [10]
3 years ago
14

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

Mathematics
1 answer:
Alex Ar [27]3 years ago
4 0

Answer:

-2

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

<u>Calculus</u>

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:

<u>Step 1: Define</u>

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

<u>Step 2: Differentiate</u>

  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

<u>Step 3: Evaluate</u>

  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
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