Calculus I Notes, Section 2-1

Notes, Lesson 2.1
The Tangent and Velocity Problems

	In this graph, we see a generic function plotted on a coordinate plane.
	We begin by taking any two points [, and ], and constructing a secant line containing them.
	We know how to calculate the slope of the secant line The slope of any secant line represents the average rate of change of the function.
Another way of referring to the is . As the point on the left gets closer and closer to the point on the right, shrinks, and the slope of the secant line containing the two points has a slope which gets closer and closer to the slope of the line which is tangent to the curve at point 2.	A derivative is the instantaneous rate of change (or slope of the tangent line) at any point on a curve. (see below)
This changing of the position of the first point is shown in the illustration below.

Another Animated secant approaching the tangent

Another animation showing the tangent line to a curve

If you have the Journey Through Calculus CD, load and run MResources/Module 1/Tangents/What is a Tangent?. This module will help you locate tangents interactively and explore them numerically.

Using your Tools for Enriching Calculus CD (that came with your book), load and run Module 2.1: Parametric Curves. This module will let you see how the process above works for live additional functions.

Check Concepts

#1: True or False: A tangent line is not only important in geometry and algebra, it is also important in the study of calculus.

#2: If the two points of intersection of a secant line and a curve continue to get closer together, then the slope of the secant gets closer and closer to the slope of the _________________.

#3: Instaneous velocity means the velocity .....

#4: Since a derivative is an instaneous rate of change or slope of the tangent line, the derivative of a distance graph would represent ....

#5 When is it that slope cannot be determined?