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sqdancefan

9514 1404 393

Answer:

  7953.873

Step-by-step explanation:

The first derivative is ...

  f'(x) = 4·3x²·e^x +4x³·e^x = e^x(4x³ +12x²)

Then the second derivative is ...

  f''(x) = (12x² +24x)e^x +(4x³ +12x²)e^x

  f''(x) = e^x(4x³ +24x² +24x)

So, f''(3) = (e^3)(4·27 +24·9 +24·3) = 396e^3 = 7953.87262158

Rounded to thousandths, this is ...

  f''(3) = 7953.873

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Answer:

f''(x) ≈ 7953.87

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

Algebra I

  • Factoring

Calculus

Derivatives

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Product Rule: [tex]\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]

Derivative Rule: [tex]\displaystyle \frac{d}{dx} [e^x]=e^x[/tex]

Step-by-step explanation:

Step 1: Define

f(x) = 4x³eˣ

f''(x) is x = 3 for 2nd Derivative

Step 2: Differentiate

  1. [1st Derivative] Product Rule [Basic Power Rule]:                                       f'(x) = 3 · 4x³⁻¹eˣ + 4x³eˣ
  2. [1st Derivative] Simplify:                                                                                 f'(x) = 12x²eˣ + 4x³eˣ
  3. [1st Derivative] Factor:                                                                                   f'(x) = eˣ(12x² + 4x³)
  4. [2nd Derivative] Product Rule [Basic Power Rule]:                                     f''(x) = eˣ(12x² + 4x³) + eˣ(2 · 12x²⁻¹ + 3 · 4x³⁻¹)
  5. [2nd Derivative] Simplify:                                                                             f''(x) = eˣ(12x² + 4x³) + eˣ(24x + 12x²)
  6. [2nd Derivative] Distribute eˣ:                                                                       f''(x) = 12eˣx² + 4eˣx³ + 24eˣx + 12eˣx²
  7. [2nd Derivative] Combine like terms:                                                           f''(x) = 24eˣx² + 4eˣx³ + 24eˣx
  8. [2nd Derivative] Factor:                                                                                 f''(x) = 4xeˣ(x² + 6x + 6)

Step 3: Evaluate

  1. Substitute:                    f''(x) = 4(3)e³(3² + 6(3) + 6)
  2. Exponents:                   f''(x) = 12e³(9 + 6(3) + 6)
  3. Multiply:                        f''(x) = 12e³(9 + 18 + 6)
  4. Add:                              f''(x) = 12e³(27 + 6)
  5. Add:                              f''(x) = 12e³(33)
  6. Multiply:                        f''(x) = 396e³
  7. Evaluate:                       f''(x) ≈ 396(20.0855)
  8. Multiply:                        f''(x) ≈ 7953.87

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