MA 242.003

• Day 21- February 5, 2013• Section 11.2: Limits and Continuity• Section 11.3: Partial Derivatives

Section 11.2: Limits and Continuity

Section 11.2: Limits and Continuity

1. Limits are at the heart of multivariable calculus

2. Understanding continuity will be fundamental for future work.

3. To show a limit DOES NOT EXIST, find two different paths into (a,b) that yield two different numbers for the limit.

3. To show a limit DOES NOT EXIST, find two different paths into (a,b) that yield two different numbers for the limit.

3. To show a limit DOES NOT EXIST, find two different paths into (a,b) that yield two different numbers for the limit.

3. To show a limit DOES NOT EXIST, find two different paths into (a,b) that yield two different numbers for the limit.

This example will be very important to us in section 11.4 on DIFFERENTIABILITY

Idea of Proof:

Idea of Proof:

Idea of Proof:

(continuation of proof)

Definition: A rational function is a ratio of two polynomials

Definition: A rational function is a ratio of two polynomials

Definition: A rational function is a ratio of two polynomials

Definition: The domain of a rational function is the set of all points where the DENOMINATOR polynomial is non-zero.

The domain of a rational function is the set of all points where the DENOMINATOR polynomial is non-zero.

The domain of a rational function is the set of all points where the DENOMINATOR polynomial is non-zero.

Types of functions we will study:

1. Polynomials:

Types of functions we will study:

1. Polynomials:

2. Rational functions:

Types of functions we will study:

1. Polynomials:

2. Rational functions:

3. Compound functions:

Types of functions we will study:

1. Polynomials: Continuous everywhere

2. Rational functions:

3. Compound functions:

Types of functions we will study:

1. Polynomials: Continuous everywhere

2. Rational functions: Continuous where defined

3. Compound functions:

Example: Find the points in space where the following rational function is continuous.

Example: Find the points in space where the following rational function is continuous.

Solution:

Example:

Proof: for a more advanced course.

Summary: Section 11.2

Summary: Section 11.2

In future work you will be required to be able to determine whether or not a function is continuous at a point.

Section 11.3: Partial Derivatives

Problem: Given a function f(x,y,z) and a point (a,b,c) in its domain, devise methods to determine the “rate of change of f in an arbitrary direction at (a,b,c)”

Section 11.3: Partial Derivatives

Problem: Given a function f(x,y,z) and a point (a,b,c) in its domain, devise methods to determine the “rate of change of f in an arbitrary direction at (a,b,c)”

Solution: Fix y=b and z = c so that f(x,b,c) is only a function of x.

Section 11.3: Partial Derivatives

Problem: Given a function f(x,y,z) and a point (a,b,c) in its domain, devise methods to determine the “rate of change of f in an arbitrary direction at (a,b,c)”

Solution: Fix y=b and z = c so that f(x,b,c) is only a function of x. Now compute the ordinary x derivative of f(x,b,c) and evaluate at x = a.

Section 11.3: Partial Derivatives

Problem: Given a function f(x,y,z) and a point (a,b,c) in its domain, devise methods to determine the “rate of change of f in an arbitrary direction at (a,b,c)”

Solution: Fix y=b and z = c so that f(x,b,c) is only a function of x. Now compute the ordinary x derivative of f(x,b,c) and evaluate at x = a. If it exists call it the x-partial derivative of f at (a,b,c)and denote it .

Section 11.3: Partial Derivatives

Problem: Given a function f(x,y,z) and a point (a,b,c) in its domain, devise methods to determine the “rate of change of f in an arbitrary direction at (a,b,c)”

Solution: Fix y=b and z = c so that f(x,b,c) is only a function of x. Now compute the ordinary x derivative of f(x,b,c) and evaluate at x = a. If it exists call it the x-partial derivative of f at (a,b,c)and denote it . Do the same for y and z.

These are very practical definitions – they tell us what to do.

These are very practical definitions – they tell us what to do.

New Notation