Help me answer this question please
2 answers:
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
Answer:
f''(x) ≈ 7953.87
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
<u>Algebra I</u>
<u>Calculus</u>
Derivatives
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Product Rule:
Derivative Rule:
Step-by-step explanation:
<u>Step 1: Define</u>
f(x) = 4x³eˣ
f''(x) is x = 3 for 2nd Derivative
<u>Step 2: Differentiate</u>
- [1st Derivative] Product Rule [Basic Power Rule]: f'(x) = 3 · 4x³⁻¹eˣ + 4x³eˣ
- [1st Derivative] Simplify: f'(x) = 12x²eˣ + 4x³eˣ
- [1st Derivative] Factor: f'(x) = eˣ(12x² + 4x³)
- [2nd Derivative] Product Rule [Basic Power Rule]: f''(x) = eˣ(12x² + 4x³) + eˣ(2 · 12x²⁻¹ + 3 · 4x³⁻¹)
- [2nd Derivative] Simplify: f''(x) = eˣ(12x² + 4x³) + eˣ(24x + 12x²)
- [2nd Derivative] Distribute eˣ: f''(x) = 12eˣx² + 4eˣx³ + 24eˣx + 12eˣx²
- [2nd Derivative] Combine like terms: f''(x) = 24eˣx² + 4eˣx³ + 24eˣx
- [2nd Derivative] Factor: f''(x) = 4xeˣ(x² + 6x + 6)
<u>Step 3: Evaluate</u>
- Substitute: f''(x) = 4(3)e³(3² + 6(3) + 6)
- Exponents: f''(x) = 12e³(9 + 6(3) + 6)
- Multiply: f''(x) = 12e³(9 + 18 + 6)
- Add: f''(x) = 12e³(27 + 6)
- Add: f''(x) = 12e³(33)
- Multiply: f''(x) = 396e³
- Evaluate: f''(x) ≈ 396(20.0855)
- Multiply: f''(x) ≈ 7953.87
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