Find the incarization of the function x) - Vx+ 4 at a-5 and use it to approximate the numbers 8.97 and y9.05. Are these approximations overestimates or underestimates? SOLUTION The derivative of fx) = (x + 4)/2 is and sa we have 5)- and 5)- Putting these values into this equation, we see that the linearicetion is L(X) - (5) - 5Xx - 5) - | + (x - 5) The corresponding linear approximation is (when x is near 5). In particular, we have V8.97 -+ (round to four decimal pleces) and V9.05 - + 13 (round to four decimal places). The lincar approximation is ilustrated in the figure to the left. We see that, indeed, the tangent line approximation is a good approximation to the given function when x is near S. We also sce that our approximations are overestimates because the tangent line lies above the ourve. Of course, a calculator could give us approximations for v8.97 and v3.05, but the linear approximation gives an approximation over an entire interval. Video Example4)