calc lesson2.2c rate of change.position TAUGHT sept02.15.notebook
September 02, 2015
Rates of change info:
Applications (a few examples): population growth rates, production rates, water flow rates, velocity, acceleration
obj: using derivatives to find rates of change and lines tangent to a function
2.2c, p116: 57­65, 83­95 all
review:
find the eqn of the tangent line to the graph at the given pt
y = x3 + x at (­1, ­2)
Today's focus: the motion of an object. Movement to the right or up is considered positive. Movement to the left or down is considered negative.
Position function: the function s that gives the position of an object as a function of time.
change in position: Δs = s(t + Δt) ­ s(t)
rate = distance/time
average velocity = change in distance/change in time = Δs/Δt
if s(t) is the position function for an object moving, then the velocity of the object at time t is Ex 9, Finding Average Velocity of a Falling Object
If a ball is dropped from a height of 100 ft, its height s at time t is given by the position function
s = ­16t2 + 100
where s is measured in ft and t is measured in seconds. Find the average velocity for each interval.
velocity is the derivative of the position function. velocity can be negative, zero or positive.
the speed of an object is the absolute value its velocity. speed cannot be negative.
the position of a free falling object
(no air­resistance and with gravity)
a. [1, 2] b. [1, 1.5] c. [1, 1.1]
s0 is initial height of object.
v0 is initial velocity.
g is acceleration due to gravity (approximately ­32 ft per sec per sec or ­9.8 meters/s/s)
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calc lesson2.2c rate of change.position TAUGHT sept02.15.notebook
September 02, 2015
ex 10, Using the Derivative to Find Velocity
At time t = 0, a diver jumps from a diving board
that is 32 ft above the water. The position of the diver is given by:
s(t) = ­16t2 + 16t + 32
where s is measured in ft and t is measured in seconds. a. When does the diver hit the water?
b. What is the diver's velocity at impact?
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calc lesson2.2c rate of change.position TAUGHT sept02.15.notebook
September 02, 2015
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calc lesson2.2c rate of change.position TAUGHT sept02.15.notebook
September 02, 2015
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