Consider the function \( y=x^{2}-3 x-9 \). Using the values \( x=7 \) and \( \Delta x=0.8 \), calculate \( \Delta y-d y \), Round your answer to three decimal places if necessary Answer How to enter your answer (opens in new window) \[ y-d y= \]

To calculate \( \Delta y \) and \( dy \) for the function \( y = x^2 - 3x - 9 \) at \( x = 7 \) with \( \Delta x = 0.8 \), we will follow these steps: 1. **Calculate \( y \) at \( x = 7 \)**: \[ y(7) = 7^2 - 3 \cdot 7 - 9 \] \[ = 49 - 21 - 9 = 19 \] 2. **Calculate \( y \) at \( x = 7 + \Delta x = 7.8 \)**: \[ y(7.8) = (7.8)^2 - 3 \cdot 7.8 - 9 \] \[ = 60.84 - 23.4 - 9 = 28.44 \] 3. **Calculate \( \Delta y \)**: \[ \Delta y = y(7.8) - y(7) = 28.44 - 19 = 9.44 \] 4. **Calculate the derivative \( \frac{dy}{dx} \)**: \[ \frac{dy}{dx} = 2x - 3 \] Evaluating at \( x = 7 \): \[ \frac{dy}{dx} \bigg|_{x=7} = 2 \cdot 7 - 3 = 14 - 3 = 11 \] 5. **Calculate \( dy \)**: \[ dy = \frac{dy}{dx} \cdot \Delta x = 11 \cdot 0.8 = 8.8 \] 6. **Calculate \( \Delta y - dy \)**: \[ \Delta y - dy = 9.44 - 8.8 = 0.64 \] Thus, the final answer is: \[ \Delta y - dy = 0.640 \] So, the answer is: \[ \Delta y - dy = 0.640 \]