MATH 1225 Lecture Notes - Lecture 5: Second Derivative, Third Derivative, Differentiable Function
28 May 2018
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Department
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Professor
Sec. 2.8 The Derivative as a Function
• In the previous section, we have the derivative of
f
at a fixed number
a
:
ax
afxf
h
afhaf
af axh
limlim
0
• Now, we want our derivative to be a function where the number
a
vary:
Given any number
x
for which this limit exists,
h
xfhxf
xf h
0
lim
. We called this new
function
f
the derivative of
f
.
• Question: What is the difference between “differentiable” and “differentiable at a point”?
• Definition: A function
f
is differentiable at
a
if
af
exists. It is differentiable on an open interval
ba,
,,, oraoraor
if it is differentiable at every number in the interval.
EX: Given the graph of
xf
, sketch
xf
• Derivative Notations:
xfDxDfxf
dx
d
dx
df
dx
dy
yxf x
And also
ax
dx
dy
or
ax
dx
dy
, which is a synonym for
af
.
The vertical bar is read “evaluate at”.
• Theorem: If
f
is differentiable at
a
, then
f
is continuous at
a
Question: If a function is continuous, is it differentiable?
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Document Summary
In the previous section, we have the derivative of f at a fixed number a : af lim. 0 h haf h af lim x a af xf ax: now, we want our derivative to be a function where the number a vary: hxf. Given any number x for which this limit exists, xf lim. We called this new h function f the derivative of f : question: what is the difference between differentiable and differentiable at a point , definition: a function f is differentiable at a if exists. It is differentiable on an open interval . Ba, aor or ora if it is differentiable at every number in the interval. Xf : derivative notations: xf y dy dx df dx d dx xf xdf. And also dy axdx or dy dx ax. The vertical bar is read evaluate at : theorem: if f is differentiable at a , then f is continuous at a.
