Page 16 - Calc for the Clueless
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Chapter 2
The Basics
Derivatives—Definition and Rules
We would like to study the word tangent. In the case of a circle, the line L 1 is tangent to the circle if it hits the
circle in one and only one place.
In the case of a general curve, we must be more careful. We wish to exclude lines like L 2. We wish to include
lines like L 3, even though, if extended, such a line would hit the curve again.
We also need to use the word secant. L 4 is secant to a circle if it hits the curve in two places.
Definition
Tangent line to a curve at the point P.
A. Take point P on the curve.
B. Take point Q 1 on the curve.
C. Draw PQ 1.
D. Take Q 2, Q 3, Q 4,. ., drawing PQ 2, PQ 3, PQ 4,... with Q's approaching P.
E. Do the same thing on the other side of P: R 1, R 2,.... such that R 1 and R 2 are approaching P.
F. If the secant lines on each side approach one line, we will say that this line is tangent to the curve at point P.
We would like to develop the idea of tangent algebraically. We will review the development of slope from
algebra.
