Textbook Problem

The tangent line to a curve at a point closely approximates the curve near the point. In fact, for x-values close enough to the point of tangency, the function and its tangent line are virtually indistinguishable. Problems 41 and 42 explore this relationship. Use each given function and the indicated point to complete the following.

(a) Write the equation of the tangent line to the curve at the indicated point.

(b) Use a graphing calculator to graph both the function and its tangent line. Be sure your graph shows the point of tangency.

(c) Repeatedly zoom in on the point of tangency. Do the function and the tangent line eventually become indistinguishable?

$f(x)=4x-x^2\text{ at }x\text{ = 5}$