Transcript SECTION 3.1 The Derivative and the Tangent Line Problem Remember what the notion of limits allows us to do .

SECTION 3.1
The Derivative and the Tangent Line Problem
Remember what the notion of limits allows us to do . . .
Tangency
Instantaneous Rate of Change
The Notion of a Derivative
Derivative
• The instantaneous rate of change of a function.
• Think “slope of the tangent line.”
Definition of the Derivative of a Function (p. 119)
The derivative of 𝑓 at 𝑥 is given by
𝑓 𝑥+∆𝑥 −𝑓(𝑥)
∆𝑥
∆𝑥→0
𝑓 ′ 𝑥 = lim
Provided the limit exists. For all 𝑥 for which this limit exists, 𝑓′
is a function of 𝑥.
Graphical Representation
So, what’s the point?
f(x)
f(x)
f(x)
f(x)
Notation and Terminology
Terminology
differentiation, differentiable, differentiable on an open interval (a,b)
Differing Notation Representing “Derivative”
′
𝒇 𝒙 ,
𝒅𝒚
,
𝒅𝒙
′
𝒚,
𝒅
𝒇(𝒙) ,
𝒅𝒙
𝑫𝒙 𝒚
Example 1 (#2b)
Estimate the slope of the graph at the points 𝑥1 , 𝑦1 and
𝑥2 , 𝑦2 .
Example 2
Find the derivative by the limit process (a.k.a. the formal
definition).
a. 𝑔 𝑥 = −3
b. 𝑓 𝑥 =
1
𝑥2
Example 3
Find an equation of the tangent line to the graph of 𝑓 at the
given point.
𝑓 𝑥 = 𝑥 2 − 3,
(2,1)
Graphs of 𝒇 and 𝒇′
𝒇′
𝒇
Graphs of 𝒇 and 𝒇′ (cont.)
𝒇′ 𝒙 = 𝟐𝒙
Alternative Form of the Derivative
𝑓 𝑥 − 𝑓(𝑐)
𝑓 𝑐 = lim
𝑥→𝑐
𝑥−𝑐
′
Example 4
Use the alternative form of the derivative.
𝑔 𝑥 =𝑥 𝑥−1 ,
𝑐=1
When is a function differentiable?
Theorem 3.1 Differentiability Implies Continuity
If 𝑓 is differentiable at 𝑥 = 𝑐, then 𝑓 is continuous at 𝑥 = 𝑐.
• Functions are not differentiable . . .
• at sharp turns (v’s in the function),
• when the tangent line is vertical, and
• where a function is discontinuous.
Example 5
Describe the 𝑥-values at which 𝑓 is differentiable.
a. 𝑓 𝑥 = 𝑥 2 − 9
b. 𝑓 𝑥 =
𝑥2
𝑥 2 −4