Upload
quentin-fletcher
View
216
Download
0
Embed Size (px)
Citation preview
Differentiability
Arches National Park - Park Avenue
North Window Arch
Balanced Rock
Delicate Arch
Let’s clear up some confusion…
To find the derivative of f(x)
EX: Find if
To find the derivative at a point a of f(x)
EX: Find if
𝐥𝐢𝐦𝒉→𝟎
𝒇 (𝒙+𝒉) − 𝒇 (𝒙 )𝒉
𝐥𝐢𝐦𝒙→𝒂
𝒇 (𝒙 ) − 𝒇 (𝒂)𝒙−𝒂
𝐥𝐢𝐦𝒙→𝒂
𝒇 (𝒙 ) − 𝒇 (𝒂)𝒙−𝒂
=𝐥𝐢𝐦𝒙→𝟓
𝒙𝟐−𝟐𝟓𝒙−𝟓
=𝐥𝐢𝐦𝒙→𝟓
(𝒙+𝟓)(𝒙−𝟓)(𝒙−𝟓)
=𝐥𝐢𝐦𝒙 →𝟓
(𝒙+𝟓)=𝟏𝟎
In General!
Specific!
To be differentiable, a function must be continuous and smooth.
Derivatives will fail to exist at:
corner cusp
vertical tangent discontinuity
f x x 2
3f x x
3f x x 1, 0
1, 0
xf x
x
Two theorems:
If f has a derivative at x = a, then f is continuous at x = a.
Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.
**The converse of this theorem is false!
1
2f a
3f b
Intermediate Value Theorem for Derivatives
Between a and b, must take
on every value between and .
f 1
23
If a and b are any two points in an interval on which f is
differentiable, then takes on every value between
and .
f f a
f b
p
If the derivative of a function is its slope, then for a constant function, the derivative must be zero.
0d
cdx
example: 3y
0y
The derivative of a constant is zero.
Rules for Differentiation Introduction
If we find derivatives with the difference quotient:
2 22
0limh
x h xdx
dx h
2 2 2
0
2limh
x xh h x
h
2x
3 33
0limh
x h xdx
dx h
3 2 2 3 3
0
3 3limh
x x h xh h x
h
23x
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
(Pascal’s Triangle)
4dx
dx
4 3 2 2 3 4 4
0
4 6 4limh
x x h x h xh h x
h
34x
We observe a pattern: 2x 23x 34x 45x 56x …
1n ndx nx
dx
examples:
4f x x
34f x x
8y x
78y x
power rule
We observe a pattern: 2x 23x 34x 45x 56x …
d ducu c
dx dx
examples:
1n ndcx cnx
dx
constant multiple rule:
5 4 47 7 5 35d
x x xdx
When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.
(Each term is treated separately)
d ducu c
dx dx
constant multiple rule:
sum and difference rules:
d du dvu v
dx dx dx d du dv
u vdx dx dx
4 12y x x 34 12y x
4 22 2y x x
34 4dy
x xdx
This makes sense, because:
0limh
f x h g x h f x g x
h
0 0
lim limh h
f x h f x g x h g x
h h
Example:Find the horizontal tangents of: 4 22 2y x x
34 4dy
x xdx
Horizontal tangents occur when slope = zero.34 4 0x x
3 0x x
2 1 0x x
1 1 0x x x
0, 1, 1x
Plugging the x values into the original equation, we get:
2, 1, 1y y y
(The function is even, so we only get two horizontal tangents.)
4 22 2y x x
4 22 2y x x
2y
4 22 2y x x
2y
1y
4 22 2y x x
4 22 2y x x
First derivative (slope) is zero at:
0, 1, 1x
34 4dy
x xdx