1. 2.2 Basic Differentiation Rules and Rates of Change Chapter 2 – Larson- revised 10/12

2. Blast from the Past Find the limit (if it exists) 1. 2. 3. 4.

3. Blast from the Past Find the limit (if it exists) 1. 2. 3. 4.

4. If the derivative of a function is its slope, then for a constant function, the derivative must be zero. Example: The derivative of a constant is zero. The Power Rule states that if n is a rational number, then the function is differentiable and

5. Examples: Determine the derivative of the following 1.3. 2.4. Power Rule Constant Multiple Rule Sum and Difference Rules

6. Constant Multiple Rule Sum and Difference Rules Examples: Determine the derivative of the following 1. 2. Examples: Determine the derivative of the following 3. 4.

7. Derivative of Sine and Cosine Examples: Determine the derivative of the following 1. 2.

8. Rates of Change The derivative can also be used to determine the rate of change of one variable with respect to another. One application is motion. Average velocity change in position versus change in time Example: If a ball is dropped from the top of a building (neglect air resistance), the height s of the ball in feet at time t is given by. Find the average velocity of the ball in the interval

9. Rates of Change The (instantaneous) velocity of an object is which is found by showing that the velocity function is the derivative of the position function. Example: Given find the velocity of the ball when Note: Velocity can be negative, zero, or positive. The speed of an object is the absolute value of its velocity.

10. Example: Motion 1.Find the average velocity of the given position function. 2.Compare this average rate to the instantaneous rate of change at the endpoints of the interval.

11. Example: Linear Approximation 1.Givenuse the equation of the line tangent to at to approximate the value of

12. Review: average slope: slope at a point: average velocity: instantaneous velocity: If is the position function: These are often mixed up by Calculus students! So are these! velocity = slope 

13. Intermediate Value Theorem for Derivatives Between a and b, must take on every value between and. If a and b are any two points in an interval on which f is differentiable, then takes on every value between and. 