Applications of Derivatives

• Extreme Values

• Monotonic Functions and First derivative test

• Concavity and curve sketching!!!

Overview

Extreme values of Function

Example

Local Max and Min

Finding Extrema

Critical Points

Finding Absolute Extrema

Example

Example

Exercise 4.1

Problems from Exercise 4.1

Mean Value Theorem

Roll’s Theorem

Mean Value Theorem

Examples

Mathematical Consequences

Example

Using mean value theorem find value of C?

Problem 1—4 of Exercise 4.2

First Derivative Test for local extreme values

First Derivative test for local extreme

Example

Exercise 3.3

Problems from Exercise 4.3

Graphing with y’ and y’’(Curve Sketching)

Concavity

Second Derivative Test

Example

Point of Inflection

Point of inflection may or may not exist

Second Derivative Test for Local Extreme

Example

Strategy for Graphing

Assignment # 4

Problems from Exercise 4.3 (1 – 21 odd)

Problems from Exercise 4.4 (9 – 15 odd)

Example

Stronger Form

Example

Caution !

Example (Wrong use of L Hospital Rule)

L Hospital Rule for one sided Limits

Indeterminate Forms

Example

Graphs (Trig Functions)

Example

Example

Ex 4.6

Problems:1 – 25 (Odd)

Antiderivatives

Antiderivative

Example

Example

Example (using Table)

Indefinite Integral

Example

Exercise

Solve problems from Ex 4.8(Try 1-5,17-24, 55-60)

This is just revision of Intermediate level material, try to do it yourself !!!

Example

Solution (Upper Sum)

Lower Sum

Mid point Rule

Summary

Sigma Notation

Examples

Example

Algebra Rules for Summation

Some useful Formulae

Riemann Sum

Norm of partition P

Example