Velocity and Other Rates of Change Chapter 3.4

Instantaneous Rates of Change 2 *

Instantaneous Rates of Change By “understanding instantaneous rate of change conceptually” is meant that you are able to understand and interpret The derivative of a function analytically (i.e., when you take a derivative) The derivative of a function from its graph (or the function from a graph of its derivative) The derivative of a function from a table of data The derivative of a function from a verbal description of the data 3

Instantaneous Rate of Change 4

Instantaneous Velocity 5

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Example 2: Vertical Motion 8

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Speed Velocity is a vector quantity, meaning that it has both magnitude and direction When movement is either vertical or horizontal (with respect to some axes), then velocity is either positive or negative With our usual axes, velocity is positive when movement is upward and negative when movement is downward Horizontally, velocity is positive when movement is to the right and negative when movement is to the left The speed of an object is the value of the velocity without regard to direction 10

Speed 11

Example 3: Reading a Velocity Graph A student walks around in front of a motion detector that records her velocity at 1-second intervals for 36 seconds. She stores the data in her graphing calculator and uses it to generate the time-velocity graph shown below. Describe her motion as a function of time by reading the velocity graph. When is her speed a maximum? 12

Example 3: Reading a Velocity Graph She walks forward (away from the detector) for the first 14 seconds, moves backward for the next 12 seconds, stands still for 6 seconds, and then moves forward again. Her maximum speed occurs at about 20 seconds, while walking backward. 13

Acceleration 14

Acceleration Due to Gravity 15

Example 4: Modeling Vertical Motion 16

Example 4: Modeling Vertical Motion 17

Example 4: Modeling Vertical Motion 18

Example 4: Modeling Vertical Motion 19

Example 4: Modeling Vertical Motion 20

Example 5: Studying Particle Motion 21

Example 5: Studying Particle Motion 22

Example 5: Studying Particle Motion 23

Example 5: Studying Particle Motion 24

Example 5: Studying Particle Motion 25

Example 5: Studying Particle Motion 26

Example 1: Enlarging Circles 27

Example 1: Enlarging Circles 28

Example 1: Enlarging Circles 29

Example 1: Enlarging Circles 30

Example 1: Enlarging Circles 31

Example 1: Enlarging Circles 32

Derivatives in Economics 33

Derivatives in Economics 34

Derivatives in Economics 35

Example 7: Derivatives in Economics 36

Example 7: Derivatives in Economics 37

Linear Density 38

Linear Density 39

Example 8: Linear Density 40

Example 8: Linear Density 41

Exercise