2 1 Derivative
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Transcript 2 1 Derivative
2-1: The Derivative
Objectives:
Assignment:
1. To apply the limit
definition of derivative
β’ P. 104: 11-27 odd, 34, 3741
2. To determine where the
derivative fails to exist
β’ P. 105: 81-88
3. To understand the
relationship between
differentiability and
continuity
β’ P. 105: 92, 95, 100-102
Warm Up 1
Use local linearity to
approximate the
slope of π¦ = π₯ at
π₯ = 0.
Objective 1
You will be able to apply the
limit definition of derivative
Definition of the Derivative
The derivative of π at π₯ is given by
π π₯ + βπ₯ β π(π₯)
π π₯ = lim
βπ₯β0
βπ₯
β²
provide the limit exists. For all π₯ for which
this limit exits, πβ² is a function of π₯.
Read βπ prime of π₯β
Exercise 1
Explain what is meant by π β² 3 = β1.
Notation and Stuff
Common notation for the derivative of a function:
Lagrange
πβ²(π₯)
Leibniz
π¦β²
ππ¦
ππ₯
π
π π₯
ππ₯
Euler
π·π₯ π¦
Newton
π¦
βπ primeβ
βπ¦ primeβ
βππ¦ β ππ₯β
βthe derivative of π¦ with respect to π₯β
βthe derivative of π with respect to π₯β
Fluxion:
Derivative with
respect to time
Differentiation
Differentiation
is the operation
defined by
taking the
derivative of a
function.
To differentiate means
to take the derivative
of a function.
A function is
differentiable at
a point if its
derivative exists
at that point.
A function is
differentiable
on an open
interval (π, π) if
its derivative
exists at every
point in the
interval.
Exercise 2
Use the limit definition to find the derivative
of π(π₯) = π₯ 3 + 2π₯.
Exercise 3
Find πβ²(π₯) for π(π₯) = π₯.
Exercise 4
Find the derivative with respect to π‘ for π¦ =
2
.
π‘
You will be able to determine
where the derivative fails to exist
Objective 2
Derivative Exploration
Use the Geometerβs
Sketchpad demonstration
to determine where a
function fails to be
differentiable. Youβre
looking for 3 cases. In
the demo, only 3 of the
functions are
differentiable everywhere.
Local Linearity
A tangent line at a particular point is a local linear
approximation of the function at that point. Use a
graphing calculator to find the local linear
approximation of each function at π₯ = 0.
1. π¦ = π₯
2. π¦ = π₯
3. π¦ = π₯
Where Derivatives Fear to Tread
The derivative of a function will fail to exist
under the following conditions:
π¦= π₯
Sharp Turns
π¦= π₯
π¦= π₯
Vertical
Tangent Lines
Discontinuities
Exercise 5
Find the π₯-values at which π is differentiable.
1. π π₯ = π₯ β 5
2. π π₯ =
π₯2
π₯ 2 β1
Objective 3
You will be able
to understand the
relationship
between
differentiability
and continuity
Derivative at a Point
The derivative of π at
π₯ = π is
πβ²
π π₯ β π(π)
π = lim
π₯βπ
π₯βπ
This implies that the
The derivative from the left
limβ
π₯βπ
π π₯ β π(π)
π₯βπ
=
provided this limit exists.
The derivative from the right
lim+
π₯βπ
π π₯ β π(π)
π₯βπ
Derivatives on Closed Intervals
This implies that the
π is differentiable
on the closed
interval π, π if it is
differentiable on the
open interval (π, π)
and if the derivative
from the right at π
and the derivative
from the left at π
both exist.
The derivative from the left
limβ
π₯βπ
π π₯ β π(π)
π₯βπ
=
The derivative from the right
lim+
π₯βπ
π π₯ β π(π)
π₯βπ
Exercise 6
Find the derivative from the left and right for
each function below at π₯ = 0.
1. π π₯ = π₯
2. π π₯ = π₯ 1/3
Exercise 7
Find the intervals over which each function is
differentiable and continuous.
1. π π₯ = π₯ 2
2. π π₯ = π₯
Differentiability Implies Continuity
If π is differentiable at π₯ = π, then it
is continuous at π₯ = π.
Exercise 8
Write the converse and contrapositive of the
theorem below. Determine the truth value of
each statement.
If π is differentiable at π₯ = π, then it is
continuous at π₯ = π.
2-1: The Derivative
Objectives:
Assignment:
1. To apply the limit
definition of derivative
β’ P. 104: 11-27 odd, 34, 3741
2. To determine where the
derivative fails to exist
β’ P. 105: 81-88
3. To understand the
relationship between
differentiability and
continuity
β’ P. 105: 92, 95, 100-102