What is Calculus? What is Calculus? Using Distance, Speed and Acceleration to Understand the Very Very Basics.

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Presentation transcript:

What is Calculus? What is Calculus? Using Distance, Speed and Acceleration to Understand the Very Very Basics

Example 1 – Constant Speed TimeDistanceSpeedAcceleration

Example 2 – Speeding up (constant acceleration) TimeDistanceSpeedAcceleration

Example 3 – Speeding up/slowing down (variable acceleration). TimeDistanceSpeedAcceleration

Example 4 – More Complicated Changes TimeDistanceSpeedAcceleration

From Numbers to Functions! D(t) = distance traveled V(t) = speed = change in D(t)/unit of time A(t) = acceleration = change in V(t)/unit of time = change in the change in D(t)/unit of time!

BIG CONCEPT!! One of the BIG concepts of calculus is the RELATIONSHIP between these functions. Examples If you know distance, can you find speed? If you know speed, can you find acceleration? If you know speed, can you find distance? If you know acceleration, can you find distance?

These type of situations occur in many disciplines including physical and biological sciences, economics, business and engineering. Much of Calculus (Rules!) involve these relationships between: F(x) – the quantity of something at x F’(x) – the CHANGE in the quantity at time x F’’(x) – the CHANGE IN THE CHANGE in the quantity at time x. “Distance/Speed/Acceleration”

IPOD Collection Suppose the number of songs on your IPOD as a function of the time since you have owned the IPOD is: S = 200M + 25 Says you add (change) by 200 songs per month and you started with 25 (freebies?) So your rate of change is 200 songs/month. And your CHANGE in the CHANGE is 0.

IPOD Collection (more realistic?) But what if a better model for your collection would be S = 30M M + 25 What is your rate of change after month 1? Month 2? Does your rate of change change? Is the CHANGE in the CHANGE of your song collection 0? Can we find these answers exactly?

YES WE CAN!!!!! And we have many functions to work with!