A central concept in calculus is the extension of the tangent line to a circle to an arb...

A central concept in calculus is the extension of the tangent line to a circle to an arbitrary curve. In this exercise you will investigate finding the tangent line to a circle using geometry and essentially using calculus.

a. Find the equation of the tangent line to the unit circle at the point by finding theslope of the line that coincides with the radius through the point as shown in the figure.

b. The slope of a tangent line can also be computed using the difference quotient.

• Write the upper half semi-circle as y = f (x).

• Write the difference quotient for f (x).

• Simplify the difference quotient using

• What happens to the simplified difference quotient as h → 0? Compare this result with what you found in part (a).