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Rates of change



Average Rate of Change: average velocity (throughout this site we will mainly be dealing with distance, speed and acceleration); average growth rate = slope

Rate is the ratio of two different quantities (ex: m/s, cm3/min,...etc.)


Example #1:

m(slope) = d/t = (16 - 4)/(3 - 0) = 12/3 = 4 m/s = average velocity in 0s - 3s.

Example #2:

m(slope) = 4 m/s = average velocity




Instantaneous Rate of Change: while average growth rate is the slope between two given points on a graph, instantaneous growth rate is at an instant, ie: at a point.


mAB = average growth rate = msecant (avr. velocity)
mCT = instantaneous growth rate = mTangent (inst. velocity) => slope of curve at point C.

Example: s(t) = -t2 + 9t + 1

Find the instantaneous growth rate (velocity) when:

1. t = 0s (initial velocity)

2. t = 4.5s

3. t = ?, s = 0m

Ex: At t= 2s:
mTan = y/x = 0/0 = undefined = (preferred) indeterminate

We can't use this equation as shown above. Since we are looking for the instantaneous rate of change, then the change in both x and y equals 0. To find an approximation allow the change (x and y) to approach 0.


As we let x and y approach 0, we see that the slope of the secant approaches 5. Therefore the slope of the tangent approaches 5.

Using this method find out what the instantaneous growth rate equals for the other three times indicated. Your final answers should be: 1. mTan approaches 9; 2. mTan approaches 0; 3. s = 0 when t = 9.11 and t = -0.11.



Here's a few helpful links:
Average Rate of Change
Instantaneous vs. Average rate of change
Rates of Change