mat 1221 survey of calculus section 2.1 the derivative and the slope of a graph
TRANSCRIPT
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MAT 1221Survey of Calculus
Section 2.1 The Derivative and the Slope of a Graph
http://myhome.spu.edu/lauw
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Expectations
Use pencils Use “=“ signs and “lim” notation correctly Do not “cross out” expressions doing
cancelations Do not skip steps – points are assigned to all
essential steps Start your solutions with
limx
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Reminder
WebAssign Homework 2.1 Quiz 02 on Monday Read the next section on the schedule
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Recall: What do we care?
How fast “things” are going• The velocity of a particle
• The “speed” of formation of chemicals
• The rate of change of a population
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Recall: Slope of Tangent Line
Volume
30 /slope ml s
30 /ml s
time
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Preview
Definition of Tangent Lines Definition of Derivatives The limit of Difference Quotients are the
Derivatives
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Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y Slope=?
1 3
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Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
We are going to use an “limiting” process to “guess” the slope of the tangent line at x=1.
Slope=?
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Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
First we compute the slope of the secant line between x=1 and x=3.
Slope=?
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Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
Then we compute the slope of the secant line between x=1 and x=2.
Slope=?
2
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Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
As the point on the right hand side of x=1 getting closer and closer to x=1, the slope of the secant line is getting closer and closer to the slope of the tangent line at x=1.
Slope=?
2
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Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
First we compute the slope of the secant line between x=1 and x=3.
Slope=?
2 2
(3) (1) (3) (1)Slope
3 1 2
0.5 3 0.5 1
22
f f f f
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Observation…
2( ) 0.5y f x x
x
y
1 3
Let h be the distance between the two points.
2 2
(3) (1) (3) (1)Slope
3 1 2
0.5 3 0.5 1
22
f f f f
2
( ) )1 1(hf f
h
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Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
Let us record the results in a table.
h
h slope2 21
0.10.01
( ) )1(1f fh
h
slope
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Example 1 The Tangent Problem
2( ) 0.5y f x x
x
y
1 3
We “see” from the table that the slope of the tangent line at x=1 should be _________.
h
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Use of Limit Notations
When h is approaching 0, is approaching 1.
We say as h0,
Or,
(1 ) (1)1
f h f
h
0
(1 ) (1)lim 1h
f h f
h
(1 ) (1)f h f
h
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Definition (Geometric Property)
For the graph of , the slope of the tangent line at a point x is
if it exists.
0
( ) ( )limh
f x h f x
h
( )y f x
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Definition (Function Property)
For a function , the derivative of f is
if it exists. (f is differentiable at x)
0
( ) ( )limh
f x h f xf x
h
( )y f x
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Example 2
Find the slope of the tangent line of
at x=2
2y x
2( )f x x
Tangent line at 2
?
x
slope
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Example 2
Find the slope of the tangent line of
at x=2 2( )f x x
Tangent line at 2
(2)
x
slope f
2y x
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Example 2
Find the slope of the tangent line of
at x=2
0
:
( ) ( )1. Simplifed the difference quotient
( ) ( )2. Find ( ) lim
3. Find (2)
h
Steps
f x h f x
hf x h f x
f xh
f
2y x
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Example 2 Step 1
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Example 2 Step 2
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Example 2 Step 3
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Example 2
Find the slope of the tangent line of
at x=2
2( )f x x
2( )f x xTangent line at 2
(2)
4
x
slope f
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Example 3
Find the equation of the tangent line of
at x=2
2( )f x x
2( )f x xTangent line at 2
(2)
4
x
slope f
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Recall: Point-Slope Form
The equation of a line pass through with slope m is given by
)( axmby
),( ba
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Example 3
The equation of a line pass through with slope m is given by
)( axmby
),( ba
2(2)f
(2) 4f
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Example 3
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Expectations
Use pencils Use “=“ signs and “lim” notation correctly Do not “cross out” expressions when
doing cancelations If you choose not to follow the
expectations, you paper will not be counted