MAT 1234
Calculus I
Section 3.1
Definition of the Derivative
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HW and ...
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WebAssign 3.1
What do we care?
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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
Recall
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Limit of the following form is important
f ( a  h)  f ( a )
lim
h 0
h
Geometrically, for the graph 𝑦 = 𝑓(𝑥),
the limit represents the slope of the
tangent line at 𝑥 = 𝑎
Recall
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Limit of the following form is important
f ( a  h)  f ( a )
lim
h 0
h
𝑓(𝑡) =displacement function of a particle
moving in a line at time 𝑡
The limit represents the velocity of the
particle at time 𝑡 = 𝑎
So…in a chemical reaction
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Limit of the following form is important
f ( a  h)  f ( a )
lim
h 0
h
𝑓(𝑡) = amount of a chemical formed at
time 𝑡
The limit represents how fast the
chemical is formed - the rate of change
of the amount of chemical at time 𝑡 = 𝑎
Definition (rephrase)
Let 𝑦 = 𝑓(𝑡) represents certain physical
quantity, the (instantaneous) rate of
change of that physical quantities at 𝑡 = 𝑎 is
f ( a  h)  f ( a )
lim
h 0
h
if it exists.
(This represents how fast the quantity is
changing.)
Definition (New Notation)
The derivative* of a function 𝑓 at 𝑥 = 𝑎 is
defined as
f (a )  lim
h 0
f ( a  h)  f ( a )
h
(*Introduced in Lab 3)
Example 1
Compare the values of 𝑓’(−1), 𝑓’(0), 𝑓’(1), 𝑓’(2), 𝑓’(3).
𝑦
-1
𝑦 = 𝑓(𝑥)
0
1
2
3
𝑥
Example 1
Compare the slopes of the tangent lines at 𝑥 = −1,0,1,2,3
𝑦
-1

𝑦 = 𝑓(𝑥)
0

1
2

3
𝑥

Example 2
Find the equation of the tangent line of 𝑦 =
𝑥 2 at 𝑥 = 3.
f ( x)  x 2
Tangent line at x  3
slope  ?
Example 2
Find the equation of the tangent line of 𝑦 =
𝑥 2 at 𝑥 = 3.
f ( x)  x 2
Tangent line at x  3
slope  f   3
Recall: Point-Slope Form
The equation of a line pass through (𝑎, 𝑏)
with slope 𝑚 is given by
y  b  m( x  a )
3
f (3)
f   3  lim
h 0
f (3  h)  f (3)
h
Example 2
y

 x  3
Expectations
y

 x  3
Example 3
The displacement (in meters) of a particle
moving in a straight line is given by
s(t )  3t  6t  1
3
where 𝑡 is measured in seconds. Find the
velocity of the particle at 𝑡 = 2.
Example 3  a  b   a  3a b  3ab  b
3
s (2  h)  s (2)
h 0
h
1
 lim  s (2  h)  s(2) 
h 0 h
v(2)  lim
3
2
2
3
s (t )  3t 3  6t  1

m/s
Expectations
s (2  h)  s (2)
h 0
h
1
 lim  s (2  h)  s(2) 
h 0 h
v(2)  lim

m/s
Thoughts on Quiz #2
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Very similar to a problem in your HW.
35 out of 37 got the answer correct for
that problem.
Only 15 of you know how to do it in the
quiz.
Why the difference?
Am I surprised?
Stay positive!