2.2 day 3.notebook
October 15, 2012
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2.2 day 3.notebook
October 15, 2012
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Average Rate of Change over an interval
2.2 Rates of Change
The average rate of change of a function over an interval is simply the slope of the secant line over the interval.
The average rate of change for f(x) over [A,B] is:
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2.2 day 3.notebook
October 15, 2012
Instantaneous Rate of Change
The instantaneous rate of change of a function at x=a is simply the slope tangent line at x=a ­> f'(a)
Find the average rate of change of the function over the indicated interval. Compare this average of change with the instantaneous change at the endpoints of the interval.
g(t) = 2t3 ­ 1 [0,1]
The instantaneous rate of change for f(x) at x=a is:
f'(a)
The instantaneous rate of change for f(x) at x=b is:
f'(b)
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Consider a graph of displacement (distance traveled) vs. time.
Average velocity can be found by taking:
The speedometer in your car does not measure average velocity, but instantaneous velocity.
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Velocity
• v(t)=s’(t)
• Negative velocity-going down or backwards
• Positive velocity-going forwards or up
Position Function
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2.2 day 3.notebook
October 15, 2012
Ex. 1 Finding Average Velocity of a Falling Object
• If a billiard ball is dropped from a height of 100 feet, its height s at
time t is given by the position function: s(t)=-16t2+100 where s is
measured in feet and t is measured in seconds.
• Find the average velocity over the interval [1,2]
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Ex. 2
At time t=0, a diver jumps from a platform diving board that is 32 feet above
the water. The position of the diver is given by
• s(t)=-16t2+16t+32
where s is measured in feet and t is measured in seconds.
• When does the diver hit the water?
• What is the divers velocity at impact?
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