SECTION 3.1 The Derivative and the Tangent Line Problem Remember what the notion of limits allows us to do .
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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
β’ The instantaneous rate of change of a function.
β’ Think βslope of the tangent line.β
Definition of the Derivative of a Function (p. 119)
The derivative of π at π₯ is given by
π π₯+βπ₯ βπ(π₯)
βπ₯
βπ₯β0
π β² π₯ = lim
Provided the limit exists. For all π₯ for which this limit exists, πβ²
is a function of π₯.
Graphical Representation
So, whatβs 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 βDerivativeβ
β²
π π ,
π
π
,
π
π
β²
π,
π
π(π) ,
π
π
π«π π
Example 1 (#2b)
Estimate the slope of the graph at the points π₯1 , π¦1 and
π₯2 , π¦2 .
Example 2
Find the derivative by the limit process (a.k.a. the formal
definition).
a. π π₯ = β3
b. π π₯ =
1
π₯2
Example 3
Find an equation of the tangent line to the graph of π at the
given point.
π π₯ = π₯ 2 β 3,
(2,1)
Graphs of π and πβ²
πβ²
π
Graphs of π and πβ² (cont.)
πβ² π = ππ
Alternative Form of the Derivative
π π₯ β π(π)
π π = lim
π₯βπ
π₯βπ
β²
Example 4
Use the alternative form of the derivative.
π π₯ =π₯ π₯β1 ,
π=1
When is a function differentiable?
Theorem 3.1 Differentiability Implies Continuity
If π is differentiable at π₯ = π, then π is continuous at π₯ = π.
β’ Functions are not differentiable . . .
β’ at sharp turns (vβs in the function),
β’ when the tangent line is vertical, and
β’ where a function is discontinuous.
Example 5
Describe the π₯-values at which π is differentiable.
a. π π₯ = π₯ 2 β 9
b. π π₯ =
π₯2
π₯ 2 β4