2.2 Basic Differentiation Rules and Rates of Change

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2.2 Basic Differentiation Rules and Rates of
Change
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After this lesson, you should be able to

• Find the derivative using the Constant Rule.
• Find the derivative using the Power Rule.
• Find the derivative using the Constant Multiple
  Rule and the Sum and Difference Rules.
• Find the derivative of sine and cosine function.
• Use derivatives to find rates of change.

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Slope of Tangent Line at (x, f (x))
Q
P
Let
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Rules For Computing Derivatives
Theorem 2.2
Theorem 2.3
Constant Rule
Power Rule
Ex
Ex
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Proof of Theorem 2.3
I suppose that you know the Binomial Theorem in
Algebra II. It indicates that
So,
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Proof of Theorem 2.3
I suppose that you know the Binomial Theorem in
Algebra II. It indicates that
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The Generalization of Theorem 2.3
Ex
Try
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Theorem 2.4 The Constant Multiple Rules
Ex
Proof of Theorem 2.4
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Theorem 2.5 The Sum and Difference Rules
Ex
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Proof of Theorem 2.5
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Theorem 2.6 Derivatives of Sine and Cosine
Functions
Ex
Ex
Try
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Proof of Theorem 2.6
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Examples
Example
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Examples
Example Find the derivative of
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Examples
Example Find the derivative of
Example Find the horizontal tangent line(s)
for the function
They may not appear to be horizontal tangents on
the graph, but algebraically they exist!!
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Examples
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Finding the Equation of a Tangent Line
Example Find the equation of the tangent line at
x 1 and take a peek at the graph.
peek at the graph
Pt f (1) 2,
(1, 2)
Slope f (x)
5x4 6x
f (1)
1
Equation
y 2 ( 1)(x 1)
y x 3
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Finding an Equation of a Horizontal Tangent Line
Example Find an equation for the horizontal
tangent line to
Taking the derivative and set to ZERO
peek at the graph
The tangent point is f (3) 6, (3, 6)
The equation of horizontal tangent line is y 6
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Rates of Change
Some Examples population growth rates,
production rates, velocity and acceleration
Common example is the motion of an object along a
straight line.
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Motion Along a Straight Line
s position
s(t) position function
s (ft)
s(t?t)
Ave. velocity during the time interval from t to
(t ?t)
s(t)
t (sec)
t
(t ?t)
Instantaneous velocity at time t
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Motion Along a Straight Line
v(t) gt 0,
v(t) lt 0,
v(t) 0,
stopped instantaneously
rate of change of position
s(t) position (ft)
rate of change of velocity
v(t) s (t) velocity (ft/sec)
a(t) v (t) s (t) accel (ft/s2)
speed v(t)
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Free Falling Object
gravity
initial velocity
initial position
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Example of Free Falling Object
Example A ball is thrown straight down from the
top of a 220-foot building with an initial
velocity of 22 feet/second. What is its
velocity after 3 seconds? What is its velocity
after falling 108 feet?
220
Describes the motion of the object
0
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Example of Free Falling Object
Example A ball is thrown straight down from the
top of a 220-foot building with an initial
velocity of 22 feet/second. What is its
velocity after 3 seconds? What is its velocity
after falling 108 feet?
220
Describes the motion of the object
0
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Example of Free Falling Object
Example A ball is thrown straight up from the
top of a 320-foot building with an initial
velocity of 16 feet/second. When can it reach to
the highest position? How high is it? What is
the total distance the ball traveled when it
reaches the ground? What is the velocity of the
ball when it reaches the ground?
320
Describes the motion of the object
0
ft
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Example of Free Falling Object
Example A ball is thrown straight up from the
top of a 320-foot building with an initial
velocity of 16 feet/second. When can it reach to
the highest position? How high is it? What is
the total distance the ball traveled when it
reaches the ground? What is the velocity of the
ball when it reaches the ground?
320
Describes the motion of the object
0
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Homework
Section 2.2 page 113 1-51 odd, 53ab, 55ab, 57,
61, 81-86 all
Rate of change problems 92 and 93