differentiability and rates of change. to be differentiable, a function must be continuous and...
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Differentiability and Rates of Change
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
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
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
2)
1)
Recall the connection between average rate of change an instantaneous
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
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.
7)
8)