3 - 1 Chapter 3 The Derivative

3 - 2 Section 3.1 Limits

Finding Limits using Graphs:

g(x) = (x 3 - 2x 2 )/(x-2) Finding Limits using Tables:

One-Sided Limits

One-Sided Limits:

Example 1: Using the graph Example 2: Using a table

As x → 2: Left limit? Right limit? Example: “Jump” behavior

Practice 1 Suppose and Use the limit rules to find Solution:

Practice 2 Solution: Rule 4 cannot be used here, since The numerator also approaches 0 as x approaches −3, and 0/0 is meaningless. For x ≠ − 3 we can, however, simplify the function by rewriting the fraction as Now Rule 7 can be used.

Practice 3: limit as x approaches 1?

Infinite Limits When x approaches a number, f(x) approaches infinity

Example:

Practice

Limits at infinity When x approaches infinity, f(x) approaches a finite limit L

Example

Example using table

Practice 4 Solution: Here, the highest power of x is x 2, which is used to divide each term in the numerator and denominator.

Practice:

Section 3.2 Continuity

Continuous?

Checklist for continuity at a point: Follow up: Continuity on an interval

Examples

Practice 1 Find all values x = a where the function is discontinuous. Solution: This root function is discontinuous wherever the radicand is negative. There is a discontinuity when 5x + 3 < 0

Practice 2 Find all values of x where the piecewise function is discontinuous. Solution: Since each piece of this function is a polynomial, the only x-values where f might be discontinuous here are 0 and 3. We investigate at x = 0 first. From the left, where x-values are less than 0, From the right, where x-values are greater than 0

Practice 2 Continued Because the limit does not exist, so f is discontinuous at x = 0 regardless of the value of f(0). Now let us investigate at x = 3. Thus, f is continuous at x = 3.

A graph example: f is continuous at 1 but discontinuous at 3

Section 3.3 Rates of Change

Definition: Average Rate of Change of a function on an interval [a,b] (AveR.O.C) Example:

Definition: Instantaneous Rate of Change of a function at a point a: Example:

Section 3.4 Definition of the Derivative

Same thing! This is also called the difference quotient of f(x) at x=a

Definition: Derivative function of f(x)

Finding the derivative:

Practice example: Find the derivative of x 2 at x=3

Where derivative does not exist : 1) At corners 2) At vertical tangents 3) At discontinuities