3.4 Velocity, Speed, and Rates of ChangeVelocitySpeedRates of Change

No excuses, Just do the work! Calculus Date: 11/18/14 Obj: SWBAT apply the basic rules of derivatives Bell Ringer: pg 125 #47, 46 Homework Check and Requests: SM selected problems pg pg #31-51 odds IN Class: Start lecture on rate of change Section Homework: Pg 135 #1-4 Hw: pg 136 #12-21 exclude direction change Announcements: Quiz Friday Derivatives Mon. Mandatory Tutoring Periodic functions Math Team Thursdays Winners never quit Quitters never win!! If at first you don’t succeed, Try and try again!!

Distance =Displacement = Change in position is measured in both displacementdistanceand If you were to drive 10 miles eastand then 4 miles west 10 miles east 4 miles west The difference between the starting and ending positions of an object The total amount of “ground covered” by an object or the total length of its path 14 miles6 miles

position time Distance =Displacement = And the position graph would look like this: 10 miles east4 miles west 14 miles6 miles

Consider a graph of position (distance from a starting point) vs. time. time (hours) position (miles) Average velocity can be found by taking: A B The speedometer in your car does not measure average velocity, but instantaneous velocity. (The velocity at one moment in time.)

Velocity is the first derivative of position. Mr. Murphy’s fall from the 196 foot platform can be expressed with the equation: …where t is in seconds and s is measured in feet. Given the above statement, the equation for his velocity is: …which is expressed in what units? feet/second

Find Which is easy enough except…  32 what? t (seconds) v(t) (feet/second) First, let’s look at the graph of velocity. Note that like the position graph, s ( t ), the y-axis represents velocity while the x axis represents time

Acceleration feet/second 2 Since the slope of a line is based on: …or in this case: …we are now talking about: Is an expression of… which is a rate of change of velocity. Acceleration

Acceleration is the derivative of velocity. feet/second 2 Remember: If you ever get lost, what can always save you?UNITS!

Example: Free Fall Equation Speed is the absolute value of velocity. Gravitational Constants:

Wait! So what is the difference between speed and velocity? Speed is the absolute value of velocity. If the object is moving upward… …or to the right If the object is moving downward… …or to the left But speed does not indicate direction so speed will always be positive…

Wait! So what is the difference between speed and velocity? Speed is the absolute value of velocity. If an object is moving upward at a speed of 40 ft/sec… If an object is moving downward at a speed of 40 ft/sec But regardless of the direction, the speed will always be positive…

time position acc pos vel pos & increasing acc zero vel pos & constant acc neg vel pos & decreasing velocity zero acc neg vel neg & decreasing acc zero vel neg & constant acc pos vel neg & increasing acc zero, velocity zero It is important to understand the relationship between a position graph, velocity and acceleration: WAIT! How can you tell that acceleration is positive?The slope (velocity) is tilting upward so the slope (velocity) is increasing

time acc pos vel pos & increasing acc zero vel pos & constant acc neg vel pos & decreasing velocity zero acc neg vel neg & decreasing acc zero vel neg & constant acc pos vel neg & increasing acc zero, velocity zero Based on this graph, how can you use velocity and acceleration to determine when an object is speeding up or slowing down? time position

time acc pos vel pos & increasing acc zero vel pos & constant acc neg vel pos & decreasing velocity zero acc neg vel neg & decreasing acc zero vel neg & constant acc pos vel neg & increasing acc zero, velocity zero Speeding up Slowing Down time position

v and a have the same sign v and a have opposite signs So the moral of the story is… Speeding up Slowing Down

Rates of Change: Average rate of change = Instantaneous rate of change = These definitions are true for any function. ( x does not have to represent time. ) 