Chapter Two: Section Two Part 1

1
Chapter Two Section Two (Part 1)

• Basic Differentiation Rules

2
Overview

• In 2.1 we found derivatives using the formal
  limit definition.
• In this section, we will learn differentiation
  rules so that we can find a derivative without
  using the limit notation.

3
Constant Rule

• Focus on the graph of y 3. What is the slope
  of this line?
• What is the slope of any horizontal line?

4
Development of basic rules

• What is the slope at any point on the line
• y 4x 1?
• Use the definition of the derivative to find the
  formula for the slope at any point on the line
• y 2x2 1.
• Is there a quicker way to get this answer?
• Find the formula for the slope at any point on
  the line y 3x3 4x2 6x 12.

5
The rules

• Power Rule
• Constant Multiple Rule

6
Example 1

• Find for the following

7
Example 2

• What is the equation of the tangent line to the
  graph of y x6 at (2, 64)?

8
Example 3

• Use any applicable rules to find .

9
Sum difference rule
10
Example 4

• Find for the following function

11
Derivatives of Sine and Cosine Functions

• In order to generate our derivatives for sine and
  cosine, we need to focus on the following Special
  Trig limits (from Chapter 1)

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Can You Find the Derivative Formulas For
13
Example 4

• Find

14
Homework (Part 1)

• 2.2
• p.110
• 2, 4, 12, 15, 16,
• 20, 21, 26, 41, 45

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Chapter Two Section Two (Part 2)

• Rates of Change
• (Position, Velocity, Acceleration)

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Derivatives are used

• to determine the rate of change of one variable
  with respect to another.
• For example
• 
  

17
Lets Examine a Change in Position

• Graphically
• Rate of Change of Position over Time
• 
  
• 
  

S(t)
feet
seconds
t
t ?t
Average Velocity (Between 2 points)
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Example 1

• A ball is dropped from a height of 150 feet. Its
  height at time t is given by
  , where s represents ft and t represents
  seconds.
• Find the average velocity from 1,2, 1, 1.5,
  1, 1.1

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Average Versus Instantaneous Rates

• On a previous slide we defined AVERAGE VELOCITY
  (between 2 points) as
• INSTANTANEOUS VELOCITY (at one point) is defined
  as

The word average indicates the slope formula.

• Velocity at a point is the derivative of
  Position.
• Velocity tells rate and direction.
• Speed is the absolute value of velocity.

20
General Position Formula
Acceleration due to gravity Either -32 ft/sec2
or -9.8 m/sec2
Initial Velocity
Initial Position
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Example 2

• At time t 0, a diver jumps from a diving board.
  Her position is given by
• (where s is in ft and t is in seconds).
• What is the divers initial velocity?
• What is the height of the diving board?
• When does the diver hit the water?
• What is the velocity at impact?

22
Homework (part 2)

• 2.2
• p. 111 112
• 48, 51, 63, 64, 68, 69, 71