Calculus is Mathematics of Change

The concept of Limit is the Central Idea of Calculus

2.1 The tangent and Velocity Problems

Conceptual idea of limit as "close to"

Will formalize the Theory in 2.4 with

Tangent - to a circle - intersect at only one point, but a tangent to a curve intersects only once "near" the given point.

The word Tangent is from Latin tangens which means “touching”

If a car is driving on a curved road, its headlights point along the direction of the tangent line

Example:

Consider a nearby point

Use TABLE feature of calculator to find y for following value in table

x	f(x)	Slope
5
4.5
4.1
4.01
4.001
3.999
3.99
3.9
3.5
3

Guess the value of the slope of the tangent line at P(4,2).

Write the equation of the tangent line.

Sketch the curve and tangent line on the calculator

Sketch tangent lines to the function below at each of the three points.