Ninjago

Master Wu: I'm sure you're aware of the dangers of this world. The threat of Lord Garmadon and his army of serpentine. The ninja are fighting to keep this world safe. It is a noble cause, but it comes with a high price. Are you willing to pay that price?

you think he knows that we're on a date

To find the derivative of \( f(x) = 2x^2 - 3x + 1 \) at the point \( (2, 3) \) using the limit definition, follow these steps: 1. **Confirm the point lies on the function:** \[ f(2) = 2(2)^2 - 3(2) + 1 = 8 - 6 + 1 = 3 \quad \checkmark \] 2. **Apply the limit definition:** \[ f'(2) = \lim_{h \to 0} \frac{f(2+h) - f(2)}{h} \] 3. **Compute \( f(2+h) \):** \[ f(2+h) = 2(2+h)^2 - 3(2+h) + 1 = 2(4 + 4h + h^2) - 6 - 3h + 1 = 8 + 8h + 2h^2 - 6 - 3h + 1 = 3 + 5h + 2h^2 \] 4. **Find the difference quotient:** \[ f(2+h) - f(2) = (3 + 5h + 2h^2) - 3 = 5h + 2h^2 \] \[ \frac{f(2+h) - f(2)}{h} = \frac{5h + 2h^2}{h} = 5 + 2h \] 5. **Take the limit as \( h \to 0 \):** \[ f'(2) = \lim_{h \to 0} (5 + 2h) = 5 \] **Final Answer:** \[ \boxed{5} \] **Verification using derivative rules:** \[ f'(x) = 4x - 3 \implies f'(2) = 4(2) - 3 = 8 - 3 = 5 \quad \checkmark \]