1. 3.4 Velocity and Other Rates of Change Objective Students will be able to use derivatives to analyze straight line motion and solve other problems involving rates of change

2. Instantaneous Rate of Change Derivatives as functions can be used to represent the rates at which things change in the world around us. Note: x does not have to represent time, and thus the word “instantaneous” is often omitted. When being asked for rate of change, it means instantaneous rate of change.

3. Some Notes x does not have to represent time, and thus the word “instantaneous” is often omitted. When being asked for rate of change, it means instantaneous rate of change. When a small change in x produces a large change in f(x), the function is sensitive to changes in x. The derivative f’(x) measures sensitivity.

4. Terminology Suppose an object is moving a line (say an s- axis) so that its position s on the line is a function of time t s=f(t)

6. Other notes Positive velocity indicates movement in the positive direction (either up or right). Negative velocity indicates movement in the negative direction (either down or left). Speed is always a positive value! – It does not indicate direction, whereas velocity does. Watch me Position, Velocity, and Acceleration

7. Additional Examples a) How high does the rocket go? We know the shape will be parabolic. The rocket will reach its’ max height when the position function changes direction. Meaning there will be a horizontal tangent line at the max height position. The slope of a horizontal tangent line is 0. The first derivative (velocity) of the position function is the slope of the horizontal tangent line. Find the velocity and set it equal to 0 (since we know the slope is 0) to get how long it takes to reach max height.