Transcript 2 1 Derivative

2-1: The Derivative
Objectives:
Assignment:
1. To apply the limit
definition of derivative
• P. 104: 11-27 odd, 34, 3741
2. To determine where the
derivative fails to exist
• P. 105: 81-88
3. To understand the
relationship between
differentiability and
continuity
• P. 105: 92, 95, 100-102
Warm Up 1
Use local linearity to
approximate the
slope of 𝑦 = 𝑥 at
𝑥 = 0.
Objective 1
You will be able to apply the
limit definition of derivative
Definition of the Derivative
The derivative of 𝑓 at 𝑥 is given by
𝑓 𝑥 + ∆𝑥 − 𝑓(𝑥)
𝑓 𝑥 = lim
∆𝑥→0
∆𝑥
′
provide the limit exists. For all 𝑥 for which
this limit exits, 𝑓′ is a function of 𝑥.
Read “𝑓 prime of 𝑥”
Exercise 1
Explain what is meant by 𝑓 ′ 3 = −1.
Notation and Stuff
Common notation for the derivative of a function:
Lagrange
𝑓′(𝑥)
Leibniz
𝑦′
𝑑𝑦
𝑑𝑥
𝑑
𝑓 𝑥
𝑑𝑥
Euler
𝐷𝑥 𝑦
Newton
𝑦
“𝑓 prime”
“𝑦 prime”
“𝑑𝑦 – 𝑑𝑥”
“the derivative of 𝑦 with respect to 𝑥”
“the derivative of 𝑓 with respect to 𝑥”
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 (𝑎, 𝑏) if
its derivative
exists at every
point in the
interval.
Exercise 2
Use the limit definition to find the derivative
of 𝑓(𝑥) = 𝑥 3 + 2𝑥.
Exercise 3
Find 𝑓′(𝑥) for 𝑓(𝑥) = 𝑥.
Exercise 4
Find the derivative with respect to 𝑡 for 𝑦 =
2
.
𝑡
You will be able to determine
where the derivative fails to exist
Objective 2
Derivative Exploration
Use the Geometer’s
Sketchpad demonstration
to determine where a
function fails to be
differentiable. You’re
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 𝑥 = 0.
1. 𝑦 = 𝑥
2. 𝑦 = 𝑥
3. 𝑦 = 𝑥
Where Derivatives Fear to Tread
The derivative of a function will fail to exist
under the following conditions:
𝑦= 𝑥
Sharp Turns
𝑦= 𝑥
𝑦= 𝑥
Vertical
Tangent Lines
Discontinuities
Exercise 5
Find the 𝑥-values at which 𝑓 is differentiable.
1. 𝑓 𝑥 = 𝑥 − 5
2. 𝑓 𝑥 =
𝑥2
𝑥 2 −1
Objective 3
You will be able
to understand the
relationship
between
differentiability
and continuity
Derivative at a Point
The derivative of 𝑓 at
𝑥 = 𝑐 is
𝑓′
𝑓 𝑥 − 𝑓(𝑐)
𝑐 = lim
𝑥→𝑐
𝑥−𝑐
This implies that the
The derivative from the left
lim−
𝑥→𝑐
𝑓 𝑥 − 𝑓(𝑐)
𝑥−𝑐
=
provided this limit exists.
The derivative from the right
lim+
𝑥→𝑐
𝑓 𝑥 − 𝑓(𝑐)
𝑥−𝑐
Derivatives on Closed Intervals
This implies that the
𝑓 is differentiable
on the closed
interval 𝑎, 𝑏 if it is
differentiable on the
open interval (𝑎, 𝑏)
and if the derivative
from the right at 𝑎
and the derivative
from the left at 𝑏
both exist.
The derivative from the left
lim−
𝑥→𝑐
𝑓 𝑥 − 𝑓(𝑐)
𝑥−𝑐
=
The derivative from the right
lim+
𝑥→𝑐
𝑓 𝑥 − 𝑓(𝑐)
𝑥−𝑐
Exercise 6
Find the derivative from the left and right for
each function below at 𝑥 = 0.
1. 𝑓 𝑥 = 𝑥
2. 𝑓 𝑥 = 𝑥 1/3
Exercise 7
Find the intervals over which each function is
differentiable and continuous.
1. 𝑓 𝑥 = 𝑥 2
2. 𝑓 𝑥 = 𝑥
Differentiability Implies Continuity
If 𝑓 is differentiable at 𝑥 = 𝑐, then it
is continuous at 𝑥 = 𝑐.
Exercise 8
Write the converse and contrapositive of the
theorem below. Determine the truth value of
each statement.
If 𝑓 is differentiable at 𝑥 = 𝑐, then it is
continuous at 𝑥 = 𝑐.
2-1: The Derivative
Objectives:
Assignment:
1. To apply the limit
definition of derivative
• P. 104: 11-27 odd, 34, 3741
2. To determine where the
derivative fails to exist
• P. 105: 81-88
3. To understand the
relationship between
differentiability and
continuity
• P. 105: 92, 95, 100-102