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Differentiability and Rates of Change
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To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at:
corner cusp
vertical tangent discontinuity(jump)
f x x 2
3f x x
3f x x 1, 0
1, 0
xf x
x
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True/False :
1) If a function is differentiable, then it must be continuous. give and example
2) If a function in continuous, then it must be differentiable.give an example
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continuous is f(x) , 4)()2()3
4)()()2
4)2()1
4)(
4)(
lim
limlim
lim
lim
2
22
2
2
xff
xfxf
f
Since
xf
xf
x
xx
x
x
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2)
1)
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Recall the connection between average rate of change an instantaneous
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Review:
average slope:y
mx
slope at a point:
0
lim h
f a h f am
h
average velocity:(slope)
ave
total distance
total timeV
instantaneous velocity: (slope at 1 point)
0
lim h
f t h f tV
h
If is the position function: f t
These are often mixed up by Calculus students!
So are these!
velocity = slope
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The slope of a curve at a point is the same as the slope of
the tangent line at that point.
If you want the normal line (perpendicular line), use
the negative reciprocal of the slope.
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7)
8)