Estimate \( \Delta f \) using the Linear Approximation and use a calculator to compute both the absolute error and the percentage error. \( f(x)=\sqrt{17+x} . \quad a=19 . \quad \Delta x=-0.1 \) \( \Delta f \approx-0.00833 \) With these calculations, we have determined that the square root of 35.9 The absolute error in Linear Approximation is: 0.00001 The absolute error in percentage terms is: 0.12

To delve into the world of linear approximation, it's fascinating to note that this method is rooted in calculus, specifically in the concept of the derivative. The linear approximation gives us a very close estimate of a function's value using the tangent line at a certain point. It's especially handy for functions that are difficult to compute directly, like square roots or exponentials, allowing us to make quick estimates without a calculator! Now, let's explore some common pitfalls while using linear approximation. A frequent mistake is not choosing a point \( a \) that is near the actual value \( x \). The closer \( a \) is to \( x \), the more accurate your approximation will be. Additionally, always remember that linear approximation works best for small changes (i.e., small \( \Delta x \)). Using it for larger changes can lead to significant inaccuracies! So, always check that \( \Delta x \) isn’t too large for the function's behavior around point \( a \).