Chapter 14 H-Outs

ADVANCED CALCULUS & ANALYTICAL

GEOMETRY (MATB 113)

CHAPTER 14:

“PARTIAL DERIVATIVES”

.:SYLLABUS CONTENTS:.

14.1 Functions of Several Variables14.2 Limits and Continuity in Higher Dimensions14.3 Partial Derivatives14.4 The Chain Rule14.5 Directional Derivatives and Gradient Vectors14.6 Tangent Planes and Differentials14.7 Extreme Values and Saddle Points14.8 Lagrange Multipliers14.9 Partial Derivatives and Constrained Variables

.

Advanced Calculus & Analytical Geometry ~ MATB 113

14.1 Functions of Several Variables

Learning Objectives:

At the end of this topic students should ;

be able to find the domains and ranges for the functions of two and three

variables.

be able to describe the domain of a function of two and three variables.

understand the terms relates to graph of two and three variables.

Functions of n Independent Variables

- Suppose D is a set of real-numbers (x1, x2, ….., xn).

- A real-valued function f on D is a rule that assigns a unique

(single) real number w = f(x1, x2, ….., xn) to each element in

D.

- The set D is the function’s domain.

- The set of w – values taken on by f is the function’s range.

- The symbol w is the dependent variables of f, and f is

said to be a function of the n independent variables x1 to

xn.

- We also call the xj ‘s the functions input variables and call

w the function’s output variable.

Partial Derivatives 2


Domain and Ranges

- In defining a function of more than one variable, we follow

the usual practice of excluding inputs that lead to complex

numbers or division by zero.

- The domain of a function is assumed to be the largest set

for which the defining rule generates real numbers, unless

the domain is otherwise specified explicitly.

- The range consists of the set of output values for the

dependent variable.

Example 14.1.1:

a) Let .

Find , and the domain of f.

b) Find the domain of :

(i)

(ii)



Functions of Two Variables

- Regions in the plane can have interior points and boundary

points.

- Closed intervals [a, b] include their boundary points.

- Open intervals (a, b) don’t include their boundary points.

- Intervals such as [a, b) are neither open nor closed.

- A point ( x0, y0) in a region (set) R in the xy-plane is an

interior point of R if it is the center of a disk of positive

radius that lies entirely in R

- A point ( x0, y0) is boundary point of R if every disk

centered at ( x0, y0) contains points that lie outside of R as

well as points that lie in R.



- A region is open if it consists entirely of interior points.

- A region is closed if it contains all its boundary points.

- A region in the plane is bounded if it lies inside a disk of

fixed radius.

(e.g triangles, rectangles, circles and disks)

- A region is unbounded if it is not bounded.

(e.g lines, planes)



Graphs, Level Curves and Contours of Function of Two

Variables

- There are two standard ways to picture the values of a

function f(x,y)

- One is to draw and label curves in the domain on which f has

a constant value.

- The other is to sketch the surface .

Level Curve: The set of points in the plane where a function

f(x,y) has a constant value .

Graph : The set of all points (x,y, f(x,y)) in space, for

f(x,y) in the domain of f.

Example 14.1.2:

Display the values of the functions in two ways:

-by sketching the surface z = f(x,y),

-by drawing an assortment of level curves in the function’s

domain.



a)

b)

Functions of Three Variables

- In the plane, the points where a function of two independent

variables has a constant value f(x,y) = c make a curve in the

function’s domain.

- In space, the points where a function of three independent

variables has a constant value f(x,y,z) = c make a surface in

the function’s domain.

Level Surface: The set of points in space where a function

of three independent variables has a constant

value .

- A point ( x0, y0, z0) in a region (set) R is an interior point of

R if it is the center of a solid ball that lies entirely in R.



- A point ( x0, y0, z0) is a boundary point of R if every sphere

centered at ( x0, y0, z0) enclose points that lie outside of R.

- A region is open if it consists entirely of interior points.

- A region is closed if it contains its entire boundary points.

Example 14.1.3:

If , sketch some level surfaces of f.



Example 14.1.4:

Given the function

(a) Find the domain and range of f. Then sketch the graph of f.

(b) Find the equation of level curve containing the point

. Sketch the level curve in two dimensional

system.



14.2 Limits and Continuity in Higher

Dimensions




variables.



Limits

Definition: (Limit of a Function of Two Variables)

We say that a function f(x,y) approaches the limit L as (x,y)

approaches f( x0, y0), and write

If, for every number , there exists a corresponding number

such that for all (x,y) in the domain of f,

whenever



Properties of Limits of Functions of Two Variables

The following rules hold if L, M, and k are real numbers and

and

1. Sum Rule :

2. Difference Rule :

3. Product Rule :

4. Constant Multiple Rule:

5. Quotient Rule : ,

6. Power Rule :

If r and s are integers with no common factors, and .

Provided is a real number.( If s is a even, we assume that

L > 0).



Example 14.2.1:

Find,

a)

b)

c)



Two-Path Test

- For a function of one variable with a jump discontinuity at, it proved that does not exist by showing that

and are not equal.

- When considering such one-sided limits, we may regard the

point on the x-axis with coordinate x as approaching the

point with coordinate a either from left or from the right,

respectively.

- The similar situation for functions of two variables is more

complicated, since in a coordinate plane there are infinite

numbers of different curves, or paths, along which (x, y) can

approach (a, b).

- However, if the limit in definition exists, then f(x, y) must

have the limit L, regardless of the path taken.

- Remember that, the two path test cannot be used to prove that a limit exists – only that a limit does not exist.



Two-Path Test for Nonexistence of a Limit

If two different paths to point P(a,b) produce two different

limiting values for f, then does not exist.

Example 14.2.2:

a) Show that does not exist.

b) Show that does not exist.

c) If , show

does not exists by evaluating this limit along the x-axis,

y-axis and along the line y = x.



Continuity

As with functions of single variables, continuity is defined in terms of limits.

Definition: (Continuous Function of Two Variables)

A function f(x,y) is continuous at the point ( x0, y0) if,

1. f is defined at ( x0, y0)

2. exists

3.

A function is continuous if it is continuous at every point of its domain.

Example 14.2.3:

At what points (x,y) or (x,y,z) in the plane are the functions continuous?

a)

b)



c)



14.3 Partial Derivatives




variables.



Partial Derivatives

- The process of differentiating a function of several variables

with respect to one of its variables while keeping the other

variable(s) fixed is called partial differentiation.

Definition: (Partial Derivatives of a Function of Two Variables)

If , then the partial derivatives of f with respect to x and y are the functions fx and fy respectively, defined by,

and

provided the limits exist.



Alternative Notation for Partial Derivatives

For , the partial derivatives fx and fy are denoted by,

and

The values of the partial derivatives of f(x,y) at the point (a, b)

are denoted by,

and

Example 14.3.1:

Find fx and fy , ifa) b)



Example 14.3.2:

Let ,

Evaluate

Example 14.3.3:

Let z be defined implicitly as a function of x and y by the

equation

Determine and .

Example 14.3.4:

Partial Derivatives of a function of three variables.

Let , determine fx, fy and fz .



Higher-Order Partial Derivatives

Given

Second-order partial derivatives

Mixed second-order partial derivatives

Differentiability Implies Continuity

If a function is differentiable at , then f is

continuous at .

Example 14.3.5:

For , determine the following



higher-order partial derivatives.

a. b. c. d.

Example 14.3.6:

Higher-order partial derivatives of a function of several variables.

By direct calculation, show that for the function .

(Note : If first, second, and third partial derivatives are

continuous, then the order of differentiation is immaterial)



14.4 The Chain Rule




variables.



Functions of Two Variables

- The Chain Rule formula for a function when

and are both differentiable functions of t is given in the

following theorem.

Theorem: (Chain Rule for Functions of Two Independent

Variables.)

If is differentiable and if , are

differentiable functions of t, then the composite

is a differentiable function of t and



Example 14.4.1:

a) Use the chain rule to find the derivative of ,

where and .

b) Let , where and . Find .



Functions of Three Variables

Theorem: (Chain Rule for Functions of Three Independent

Variables.)

If is differentiable and x, y, and z are differentiable

functions of t, then w is a differentiable function of t and



Example 14.4.2:

a) Use a chain rule to find if

,

with , and .

b) Find if

, , and z = t

What is the derivative’s value at t = 0?

c) Let , where and .

Find and .



Example 14.4.3:

A simple electrical circuit of a resistor R and an electromotive force V. At a certain instant V is 80 volts and is increasing at a rate of 5 volts/min, while R is 40 ohms and is decreasing at a rate of 2 ohms/min. Use Ohm’s law, I = V/R, and a chain rule to find the rate at which the current I (in amperes) is changing.

Functions Defined on Surfaces

Theorem: (Chain Rule for Two Independent Variables and

Three Intermediate Variables.)

Suppose that , , and . If all four

functions are differentiable, then w has partial derivatives with

respect to r and s, given by the formulas,





Example 14.4.4:

a) Express and in terms of r and s if,

, , and z = 2r

b) Find if , where ,

and .

c) If f is differentiable and , show that



Implicit Differentiation Revisited

The two-variable Chain Rule leads to a formula that takes some

of the algebra out of implicit differentiation. Suppose that

1. The function F(x,y) is differentiable and

2. The equation F(x,y) = 0 defines y implicitly as a

differentiable function of x, say y = h(x).

Since w = F(x,y) = 0, the derivative dw/ dx must be zero.

Computing the derivative from the chain rule, we find

If , we can solve this equation for dy/dx to get

.

Theorem A Formula for Implicit Differentiation

Suppose that F(x,y) is differentiable, and that the equation



F(x,y) = 0 defines y as differentiable function of x. Then at any

point where ,

Example 14.4.5:

If y is a differentiable function of x such that

Find dy/dx.

Example 14.4.6:

Find and if is determined implicitly by

14.5 Directional Derivatives and Gradient

Vectors






variables.



Directional Derivatives

- We have seen that the partial derivatives of a function give

the instantaneous rates of change of that function in

directions parallel to the coordinate axes.

- Directional derivatives allow us to compute the rates of

change of a function with respect to distance in any

direction.

- Suppose that we wish to compute the instantaneous rate of

change of a function with respect to distance from a

point (x0, y0) in some direction.



- Since there are infinitely many different directions from

(x0, y0) in which we could move, we need a convenient

method for describing a specific direction starting at (x0, y0).

- One way to do this is to use a unit vector

u = u1i + u2j

that has its initial point at (x0, y0) and points in the desired

direction.

- This vector determines a line l in the xy-plane that can be

expressed parametrically as

and

where s is the arc length parameter that has its reference

point at (x0, y0) and has positive values in the direction of u.

- For s =0, the point (x0, y0) is at the reference point (x0, y0),

and as s increases, the point (x0, y0) moves along l in the

direction of u.



Definition: (Directional Derivative)

The derivative of f at P0 (x0, y0) in the direction of the unit

vector u = u1i + u2j is the number

provided the limit exists.

- The directional derivative above is also denoted by



- Geometrically, Duf(x0, y0) can be interpreted as the slope

of the surface in the direction of u at the point

(x0, y0).

Gradient

The directional derivative Duf(x, y) can be expressed concisely

in terms of a vector function called gradient.

Definition: (Gradient)

The gradient vector (gradient) of f(x, y) at a point

is the vector (pronounced “del eff”) given by

Obtained by evaluating the partial derivatives of f at

Note: Think of the symbol as an “operator” on a function

that produces a vector. Another notation for is grad f(x, y).



Example 14.5.1:

Find the gradient of at the point (1,1)

Theorem 9: (The Directional Derivative is a Dot Product)

If is differentiable in an open region containing

, then

The dot product of the gradient at and u

Example 14.5.3:

Find the derivative of the function

at in the direction of

Example 14.5.4:

Let

(a) Find the gradient of f at the point P(1,2), and sketch the Partial Derivatives 35


vector .

(b) Use the gradient to find the directional derivative of f at

P(1,2) in the direction from P(1,2) to Q(2,5).



Properties of the Directional Derivative

1. The function f increases most rapidly when or

when u is in the direction of . That is, at each point P in

its domain, f increases most rapidly in the direction of

the gradient vector at P. The derivative in this direction is

2. Similarly, f decreases most rapidly in the direction of .

The derivative in this direction is

3. Any direction u orthogonal to a gradient is a

direction of zero change in f because then equals and

Example 14.5.5:

(Maximal Rate of Increase and Decrease)



Let

(a) Find the direction in which f(x,y) increase most rapidly at

the point P(1, 2), and find the maximum rate of increase

of f at P.

(b) Interpret (a) using the graph of f .

Example 14.5.6:

( Function of Three Variables )

Suppose an xyz-coordinate system is located in space such that te

temperature T at the point (x,y,z) is given by the formula

(a) Find the rate of change of T with respect to distance at the

point P(1,3,-2) in the direction of the vector

(b) In what direction from P to T increase most rapidly ?

What is the maximum rate of change of T at P ?



Gradients and Tangents to Level Curves

At every point in the domain of a differentiable function , the gradient of f is normal to the level curve through .

Example 14.5.7

Given .

(a) Sketch the curve together with and

the tangent line at .

(b) Then write an equation for the tangent line.

Algebra Rules for Gradients

1. Constant Multiple Rule , k any number

2. Sum Rule

3. Difference Rule

4. Product Rule

5. Quotient Rule



14.6 Tangent Planes and Differentials



be able to find the equation of the tangent plane from the partial derivatives

of the function defining the surface.

be able to find the parametric equation of the normal lines to the function

defining the surface.

understand the total differential and linearization of functions of several

variables.

Tangent Planes and Normal Lines

Definitions: (Tangent Planes and Normal Lines)

The tangent plane at the point P0 (x0, y0, z0) on the level surface

of a differentiable function f is the plane through P0

normal to .

The normal line of the surface at P0 is the line through P0

parallel to .



Equations for tangent plane and normal line :

Tangent Plane to at

Normal Line to at

____________________________________________________

Example 14.6.1:

Find the tangent plane and normal line of the surface:

(a) at the point P0 (1, 2, 3).

(b) at the point P0 (0, 1, 2).



Plane Tangent to a Surface at

The plane tangent to the surface of a

differentiable function f at the point is

Example 14.6.2:

Find an equation of the tangent plane to

at the point .

Example 14.6.3:

Find parametric equations of the tangent line to the curve of intersection of the paraboloid and the ellipsoid at the point ( 1, 1, 2).

____________________________________________________

Estimating Change in a Specific Direction



Estimating the Change in f in a Direction u

To estimate the change in the value of a differentiable function f when we move a small distance ds from a point

in a particular direction u, use the formula

distance increment Directional Derivative

How to Linearize a Function of Two Variables

where and

As and ,

Definitions

The linearization of a function at a point where is

differentiable is the function



The approximation

is the standard linear approximation of f at

Example 14.6.4

Find the linearization of the function

at

Differentials

Definition: (Differentials)

If we move from (x0, y0) to a point (x0 + dx, y0+ dy) nearby,

the resulting change

in the linearization of f is called the total differential of f.

Example 14.6.5:

Determine the total differential of the given functions:

a.

b.Partial Derivatives 44


Example 14.6.6:

Suppose that a cylindrical can is designed to have a radius of 1

inch and a height of 5 inch, but that the radius and height are off

by the amounts dr = +0.03 and dh =-0.1. Estimate the resulting

absolute change in the volume of the can.


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