chapter_14.1_-_14.4
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CalculusTRANSCRIPT
Advanced Calculus & Analytical Geometry ~ MATB 113
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
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1Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
14.1 Functions of Several VariablesLearning 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.
2Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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)
3Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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.
4Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
- 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)
5Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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.
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Advanced Calculus & Analytical Geometry ~ MATB 113
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 .
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Advanced Calculus & Analytical Geometry ~ MATB 113
- 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.
8Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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.
9Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
14.2 Limits and Continuity in Higher DimensionsLearning 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.
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
10Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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).
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Advanced Calculus & Analytical Geometry ~ MATB 113
Example 14.2.1:
Find,
a)
b)
c)
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Advanced Calculus & Analytical Geometry ~ MATB 113
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.
13Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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.
14Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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)
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Advanced Calculus & Analytical Geometry ~ MATB 113
c)
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Advanced Calculus & Analytical Geometry ~ MATB 113
14.3 Partial DerivativesLearning 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.
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.
17Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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)
18Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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 .
19Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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
20Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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)
21Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
14.4 The Chain RuleLearning 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 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
22Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
Example 14.4.1:
a) Use the chain rule to find the derivative of ,
where and .
b) Let , where and . Find .
23Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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
24Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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 .
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Advanced Calculus & Analytical Geometry ~ MATB 113
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,
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Advanced Calculus & Analytical Geometry ~ MATB 113
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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
28Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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 29
Partial Derivatives
Advanced Calculus & Analytical Geometry ~ MATB 113
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
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