Two kinds of rate of change Q: A car travels 110 miles in 2 hours. What’s its average rate of change (speed)? A: 110/2 = 55 mi/hr. That is, if we drive 55 miles in an hour, then in 2 hours, we will have driven 110 miles. Q: If you are driving and suddenly look at your odometer, which says 60 mi/hr, what kind of rate of change is that? A: Instantaneous R.O.C. That is, the rate at that particular time instance. Average R.O.C. is over a period of time Instantaneous R.O.C. is at a given point of time.
Average rate of change Height (feet) A rocket is shot straight up, given by the function f(x) = -16x^2+128x, where x = time, f(x) = height at time x. 8 P Q Time (seconds) Q: What is the average speed between P (at x = 4) and Q (at x = 7)? 1. Find the slope of the secant line between P and Q. 2. Find the coordinates of P and Q. For P, f(4) = -16(4)^2+128(4) = 256, P(4,256). For Q, f(7) = -16(7)^2+128(7) = 112, so Q(7,112). 3. Slope = ( )/(7-4) = -48 ft/sec
Instantaneous rate of change Height (feet) Q: What is the average R.O.C as x changes from 4 to 6? Q: What is the average R.O.C as x changes from 4 to 5? Q: What is the average R.O.C as x changes from 4 to 4.001? P Time (seconds) What do you think the instantaneous R.O.C would be at x = 4? How fast is the rocket moving at precisely X = 4 seconds?
Definition Average Rate of Change of a quantity over a period of time is the amount of change divided by the time required for the change.
Example time in hours distance in miles Average speed is 35.7 mph measured over 7 hours
Important Idea The average rate of change (speed) over a time period is the slope of the secant line connecting the beginning and end of the time period. Average Rate of Change
Try This Describe in words how you could find the speed at exactly the 5th hour. time in hours distance in miles 0 5
Solution The instantaneous velocity at exactly the 5 th hour is the slope of the line tangent to the velocity function at t =5.
Important Idea The instantaneous velocity at a point, or any other rate of change, is the slope of the tangent line at the point
The derivative can be used to determine the rate of change of one variable with respect to another. Ex: Population growth, production rates, rate of water flow, velocity and acceleration. Ex: Free fall Position function. A function, s, that gives position (relative to the origin) of an object as a function of time.
A ball dropped from a 160 foot building: Find average velocity over each time interval. Therefore, the average velocity is Negative velocity indicates object is falling
Find instantaneous velocity when t=1.1 sec Generally if, s = s(t) is the position for an object moving in a straight line, then the velocity of the object at time t is:
Position Function Velocity Function Acceleration Function