Estimate \( \Delta f \) using the Linear Approximation and use a calculator to compute both the absolute error and the percentage error. \( f(x)=\sqrt{17+x} . \quad a=19 . \quad \Delta x=-0.1 \) \( \Delta f \approx \square \) With these calculations, we have determined that the square root of \( \square \) is approximately The absolute error in Linear Approximation is: The absolute error in percentage terms is:

To estimate \( \Delta f \) using linear approximation, we have \( f(x) = \sqrt{17+x} \). The derivative \( f'(x) \) can be calculated as follows: \[ f'(x) = \frac{1}{2\sqrt{17+x}} \] At \( a = 19 \): \[ f'(19) = \frac{1}{2\sqrt{36}} = \frac{1}{12} \] Now, using \( \Delta x = -0.1 \): \[ \Delta f \approx f'(19) \cdot \Delta x = \frac{1}{12} \cdot (-0.1) = -\frac{1}{120} \approx -0.00833 \] Using this, we determine that \( f(19) \approx \sqrt{36} = 6 \), so: \[ f(18.9) \approx 6 + \Delta f \approx 6 - 0.00833 \approx 5.99167 \] Now compute the actual value: \[ f(18.9) = \sqrt{17 + 18.9} = \sqrt{35.9} \approx 5.9851 \] The absolute error is: \[ |f(18.9) - (6 - 0.00833)| = |5.9851 - 5.99167| \approx 0.00657 \] The percentage error is given by: \[ \left(\frac{\text{absolute error}}{\text{actual value}}\right) \times 100 \approx \left(\frac{0.00657}{5.9851}\right) \times 100 \approx 0.109\% \] In conclusion: \[ \Delta f \approx -0.00833 \] The absolute error in Linear Approximation is: \[ 0.00657 \] The absolute error in percentage terms is: \[ 0.109\% \]