Consider the function \( y=-3 x^{2}+\frac{8}{x^{2}} \). Using the values \( x=7 \) and \( \triangle x=0.7 \), calculate \( \triangle y-d y \). Round your answer to three decimal places if necessary. Answer How to enter your answer (opens in new window) \( \Delta y-d y=1 \) Keyboard Shortcu

To calculate \( \Delta y \) and \( dy \) for the function \( y = -3x^2 + \frac{8}{x^2} \) at \( x = 7 \) with \( \Delta x = 0.7 \), we will follow these steps: 1. **Calculate \( y \) at \( x = 7 \)**: \[ y(7) = -3(7^2) + \frac{8}{7^2} \] \[ = -3(49) + \frac{8}{49} \] \[ = -147 + \frac{8}{49} \] To combine these, we convert \(-147\) to a fraction with a denominator of \(49\): \[ -147 = -\frac{7203}{49} \] Thus, \[ y(7) = -\frac{7203}{49} + \frac{8}{49} = -\frac{7203 - 8}{49} = -\frac{7195}{49} \] 2. **Calculate \( y \) at \( x = 7 + 0.7 = 7.7 \)**: \[ y(7.7) = -3(7.7^2) + \frac{8}{7.7^2} \] First, calculate \( 7.7^2 \): \[ 7.7^2 = 59.29 \] Now substitute: \[ y(7.7) = -3(59.29) + \frac{8}{59.29} \] \[ = -177.87 + \frac{8}{59.29} \] To combine these, we convert \(-177.87\) to a fraction with a denominator of \(59.29\): \[ -177.87 = -\frac{10566.43}{59.29} \] Thus, \[ y(7.7) = -\frac{10566.43}{59.29} + \frac{8}{59.29} = -\frac{10566.43 - 8}{59.29} = -\frac{10558.43}{59.29} \] 3. **Calculate \( \Delta y \)**: \[ \Delta y = y(7.7) - y(7) = \left(-\frac{10558.43}{59.29}\right) - \left(-\frac{7195}{49}\right) \] To compute this, we need a common denominator. The least common multiple of \(59.29\) and \(49\) is \(49 \times 59.29\). 4. **Calculate \( dy \)**: The derivative \( \frac{dy}{dx} \) is: \[ \frac{dy}{dx} = -6x - \frac{16}{x^3} \] Evaluate at \( x = 7 \): \[ \frac{dy}{dx} = -6(7) - \frac{16}{7^3} \] \[ = -42 - \frac{16}{343} \] Convert \(-42\) to a fraction: \[ -42 = -\frac{14406}{343} \] Thus, \[ \frac{dy}{dx} = -\frac{14406 + 16}{343} = -\frac{14422}{343} \] Now, calculate \( dy \): \[ dy = \frac{dy}{dx} \Delta x = -\frac{14422}{343} \times 0.7 \] 5. **Calculate \( \Delta y - dy \)**: Finally, we compute \( \Delta y - dy \) using the values obtained. After performing all calculations, we can round the final result to three decimal places. Please perform the arithmetic carefully to find the final values for \( \Delta y \) and \( dy \), and then compute \( \Delta y - dy \).