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)