VECTOR ALGEBRA
MATH23
MULTIVARIABLE CALCULUS
GENERAL
OBJECTIVE
At the end of the lesson the students are
expected to:
Perform dot product and cross product
Apply the concepts of dot product to get the angle
between two vectors
Apply cross products in v
QUADRIC SURFACES
MATH23
MULTIVARIABLE CALCULUS
Week 2 Day 2
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Discuss quadric surfaces, its equations and graphs.
Differentiate various quadrics, based on equation, and graph
Example
QUADRIC SURFACES
MATH23
MULTIVARIABLE CALCULUS
Week 2 Day 2
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Discuss quadric surfaces, its equations and graphs.
Differentiate various quadrics, based on equation, and graph
Example
PARTIAL DERIVATIVES
MATH23
MULTIVARIABLE CALCULUS
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Determine the geometric interpretation of partial
derivatives and its derivation
PARTIAL DERIVATIVES
Geometric Interpretation of Par
MAXIMA AND MINIMA
MATH23
MULTIVARIABLE CALCULUS
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Determine the geometric interpretation of partial
derivatives and its derivation
COURSE OUTCOME 1
WEEK 3 LINES AND PLANES IN SPACE
Week 2 Day 2
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Find equation of lines using various methods.
Determine an equations of a plane and determine the vector
of its norma
PARTIAL DERIVATIVES
MATH23
MULTIVARIABLE CALCULUS
GENERAL
OBJECTIVE
At the end of the lesson the students are
expected to:
Determine the geometric interpretation of partial
derivatives and its derivation
PARTIAL DERIVATIVES
Definition
Let z = f(x,y). The
MAXIMA AND MINIMA
MATH23
MULTIVARIABLE CALCULUS
GENERAL
OBJECTIVE
At the end of the lesson the students are
expected to:
Determine the geometric interpretation of partial
derivatives and its derivation
MAXIMA AND MINIMA
Graphical Interpretation
Theorems o
LEVEL CURVES AND
LEVEL SURFACES
MATH23
MULTIVARIABLE CALCULUS
GENERAL
OBJECTIVE
At the end of the lesson the students are
expected to:
Determine graphs of the level curves of a given function
Sketch and identify the level surfaces of a function
Definiti
MULTIPLE INTEGRATION
MATH23
MULTIVARIABLE CALCULUS
GENERAL
OBJECTIVE
At the end of the lesson the students are
expected to:
Determine the geometric interpretation of partial
derivatives and its derivation
14.1.1 (p. 1001)
The Volume Problem
Figure 14.1.2
QUADRIC SURFACES
MATH23
MULTIVARIABLE CALCULUS
GENERAL
OBJECTIVE
At the end of the lesson the students are
expected to:
Determine functions of several variables.
Identify the domain of several variables
Discuss quadric surfaces, its equations and graphs.
SERIES
MATH23
MULTIVARIABLE CALCULUS
GENERAL
OBJECTIVE
At the end of the lesson the students are
expected to:
Define Series and Sequences
Determine whether an infinite series is divergent or
convergent
Test for convergence of alternating series.
Express n
VECTORS IN SPACE
MATH23-1
1ST Quarter
AY 2016-2017
Vectors in space
z
The point P( x, y, z ) has
the position vector
P( x, y, z )
v OP x, y, z
v
o
The length or magnitude of
vector v is denoted by
z
y
x
x
y
v x2 y2 z2
z
If A (a1 , a2 , a3 ) and B (b1 , b2
VECTOR-VALUED FUNCTION
MATH23-1
1ST Quarter
AY 2016-2017
When a point moves along a curve in three
dimensional space, the points changing
position can be described by means of
parametric equations
z
x f (t ) , y g (t ) , z h(t )
that specifies its coordin
LINES AND PLANES IN SPACE
MATH23-1
1ST Quarter
AY 2016-2017
Lines in space
Just in the plane, a straight line in space is determined by any two points that lie
on it.
z
L
0
v ai bj ck
r0
Then another point P( x, y, z ) lies on
the line L if and only if v
SPACE COORDINATES, PLANES
and SURFACES
CALCULUS 3
MATH23-1
1st TERM AY 2016-2017
Let OX, OY, and OZ be three mutually perpendicular
lines. These lines constitute the x-axis, the y-axis, and the
z-axis of a three-dimensional rectangular coordinate
system.
Question 1
1 out of 1 points
3
Find y implicitly from the equation 3x + 5y = x + y.
Question 2
0 out of 1 points
2
Locate the centroid of the area bounded by x-axis and y = 5 + 3x .
Question 3
0 out of 1 points
Sara is about to take a test that contains
QUADRIC SURFACES
MATH23-1
CALCULUS 3
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Discuss quadric surfaces, its equations and graphs.
Differentiate various quadrics, based on equation, and graph.
Examples:
LINES AND PLANES IN SPACE
MATH23-1
CALCULUS 3
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Find equation of lines using various methods.
Determine an equations of a plane and determine the vector
of its normal.
Examples:
MATH23X EXIT EXAM REVIEW ALGEBRA
1. Find the value of
a) -2
x in the equation
3x 2( x 2) 6
b) -1
c)
.
2
d) 4
2. A student in an algebra course has test scores of 75, 82, 71, and 84. What score on the
next test will raise the students average to 80?
a)
85
VECTOR OPERATIONS
MATH23-1
CALCULUS 3
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Perform dot product and cross product.
Apply the concepts of dot product to get the angle between
two vectors.
Apply cross products in variou
VECTOR VALUED FUNCTIONS
MATH23-1
CALCULUS 3
GENERAL OBJECTIVE
At the end of the lesson the students are expected to:
Differentiate a vector valued function, and
Integrate a vector valued function.
Definition: A vector-valued function or vector
function
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