MAT 1234 Calculus I Section 2.3 Part I Using the Limit Laws .
MAT 1234 Calculus I Section 2.1 Part II Derivatives and Rates of Change.
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Transcript of MAT 1234 Calculus I Section 2.1 Part II Derivatives and Rates of Change.
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MAT 1234Calculus I
Section 2.1 Part II
Derivatives and Rates of Change
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HW
WebAssign HW 2.1 II Be sure to read the instructions carefully. Turn in the written HW at the end of your
handout.
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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 charges in a capacitor
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Recall
Limit of the following form is important
Geometrically, for the graph , the limit represents the slope of the tangent line at
h
afhafh
)()(lim
0
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Recall
Limit of the following form is important
displacement function of a particle moving in a line at time
The limit represents the velocity of the particle at time
h
afhafh
)()(lim
0
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So…in a chemical reaction
Limit of the following form is important
amount of a chemical formed at time The limit represents how fast the
chemical is formed - the rate of change of the amount of chemical at time
h
afhafh
)()(lim
0
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Definition (rephrase)
Let represents certain physical quantity, the (instantaneous) rate of change of that physical quantities at is
if it exists.
(This represents how fast the quantity is changing.)
h
afhafh
)()(lim
0
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Definition (New Notation)
The derivative* of a function at is defined as
(*Introduced in Lab 3)
h
afhafaf
h
)()(lim)(
0
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Example 1
Compare the values of )3(),2(),1(),0(),1( fffff
𝑥
𝑦
1
𝑦= 𝑓 (𝑥 )
-1 2 30
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Example 1
Compare the slopes of the tangent lines at 3 ,2 ,1 ,0 ,1x
𝑥
𝑦
1
𝑦= 𝑓 (𝑥 )
-1 2 30
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Example 2
Suppose we model the amount of certain drug inside a patient’s body by mg after hours of injection.
)(tQ
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Example 2(a)
)(aQWhat is the meaning of ?
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Example 2(b)
mg/hour? 4)3( QWhat is the meaning of
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Example 2(c)
mg/hour? 4)3( Q
How to determine the units (mg/hour)?
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Example 3
(a) Find if
(b) Find
)(af 1)( xxf
(1), (2), and (3)f f f
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Example 3
(a) Find if)(af 1)( xxf
0
( ) ( )( ) lim
h
f a h f af a
h
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Example 3
(a) Find if
(b) Find
)(af 1)( xxf
(1), (2), and (3)f f f
(1)
(2)
(3)
f
f
f
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Remarks
1)( xxf 12
1)(
aaf
If we want to find , ,and , we do not need to compute 3 limits. We only need to substitute 1, 2, and 3 into the formula of above.
This practice treats as a function:
given , the formula gives
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Classwork Hint
#2 Do NOT expand the denominator
0 3
1li
3m
h h a h a
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Quiz
Please take a look at the grader’s comments
Some of you did really well Some of you have a lot of room to
improve !!!! Explaining your work clearly and
carefully is VERY important It is not too late to get help