MAT 1221 Survey of Calculus Section 2.1 The Derivative and the Slope of a Graph

Expectations Use pencils Use “=“ signs and “lim” notation correctly Do not “cross out” expressions doing cancelations Do not skip steps – points are assigned to all essential steps Start your solutions with

Reminder WebAssign Homework 2.1 Quiz 02 on Monday Read the next section on the schedule

Recall: What do we care? How fast “things” are going The velocity of a particle The “speed” of formation of chemicals The rate of change of a population

Recall: Slope of Tangent Line

Preview Definition of Tangent Lines Definition of Derivatives The limit of Difference Quotients are the Derivatives

Example 1 The Tangent Problem Slope=?

Example 1 The Tangent Problem We are going to use an “limiting” process to “guess” the slope of the tangent line at x=1. Slope=?

Example 1 The Tangent Problem First we compute the slope of the secant line between x=1 and x=3. Slope=?

Example 1 The Tangent Problem Then we compute the slope of the secant line between x=1 and x=2. Slope=?

Example 1 The Tangent Problem As the point on the right hand side of x=1 getting closer and closer to x=1, the slope of the secant line is getting closer and closer to the slope of the tangent line at x=1. Slope=?

Example 1 The Tangent Problem First we compute the slope of the secant line between x=1 and x=3. Slope=?

Observation… Let h be the distance between the two points.

Example 1 The Tangent Problem Let us record the results in a table. hslope

Example 1 The Tangent Problem We “see” from the table that the slope of the tangent line at x=1 should be _________.

Use of Limit Notations When h is approaching 0, is approaching 1. We say as h  0, Or,

Definition (Geometric Property) For the graph of, the slope of the tangent line at a point x is if it exists.

Definition (Function Property) For a function, the derivative of f is if it exists. ( f is differentiable at x )

Example 2 Find the slope of the tangent line of at x=2

Example 2 Find the slope of the tangent line of at x=2

Example 2 Find the slope of the tangent line of at x=2

Example 2 Step 1

Example 2 Step 2

Example 2 Step 3

Example 2 Find the slope of the tangent line of at x=2

Example 3 Find the equation of the tangent line of at x=2

Recall: Point-Slope Form The equation of a line pass through with slope m is given by

Example 3 The equation of a line pass through with slope m is given by

Example 3

Expectations Use pencils Use “=“ signs and “lim” notation correctly Do not “cross out” expressions when doing cancelations If you choose not to follow the expectations, you paper will not be counted