Applications of Derivatives

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Presentation transcript:

Applications of Derivatives Chapter 4 Applications of Derivatives

What you’ll learn about 4.1 Extreme Values of Functions Homework: pg.193 #1(ex.1), 3(ex. 2), 11 (ex.3) review: solve inequality: pg.18 quick review # 1, 2, 6 What you’ll learn about Absolute (Global) Extreme Values Local (Relative) Extreme Values Finding Extreme Values …and why Finding maximum and minimum values of a function, called optimization, is an important issue in real-world problems.

What you’ll learn about Absolute (Global) Extreme Values Local (Relative) Extreme Values Finding Extreme Values …and why Finding maximum and minimum values of a function, called optimization, is an important issue in real-world problems.

Match the table with the graph:

Example1

Example2

Example Finding Absolute Extrema Find critical points Find critical points values Endpoints values Chose max and min values which are absolute extremas

Example Finding Extreme Values

Homework: # 22, 25 pg.194 pg.18 quick review # 3, 4, 5 2/19/2013 Extreme Values of Functions Objectives: Absolute (Global) Extreme Values Local (Relative) Extreme Values Finding Extreme Values Homework: # 22, 25 pg.194 pg.18 quick review # 3, 4, 5 While a function’s extrema can occur only at critical points and endpoints, not every critical point or endpoint signals the presence of an extreme value.

Read example 4 pg. 190, solve # 27, 28 pg.194 Example 6 read together, exploration 1 (optional)

Mean Value Theorem Homework: pg.202 #1(ex.1), 4 4.2 Mean Value Theorem Homework: pg.202 #1(ex.1), 4

What you’ll learn about Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences …and why The Mean Value Theorem is an important theoretical tool to connect the average and instantaneous rates of change.

Continuous? Differentiable?

Example Explore the Mean Value Theorem

# 11-14 pg. 202

Objectives: Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Homework: pg.202 # 15, 20, 26 Objectives: Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences …and why The Mean Value Theorem is an important theoretical tool to connect the average and instantaneous rates of change.

Increasing Function, Decreasing Function

Corollary: Increasing and Decreasing Functions

Example Determining Where Graphs Rise or Fall

First Derivative Test for Local Extrema

First Derivative Test for Local Extrema

Example Using the First Derivative Test solve # 16, 22, 18, 21, 25 pg.202

4.2 Other Consequences Objectives: Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences Homework: read example 7 pg.200 solve # 35, 38, 43 (pg. 203)

Corollary: Functions with f’=0 are Constant

Corollary: Functions with the Same Derivative Differ by a Constant

Antiderivative

Example Finding Velocity and Position

Example 5 pg.,