MATH23X
EXIT EXAMINATION
2nd Quarter SY 20132014
DIRECTIONS: Solve each problem and choose the best answer from the choices given.
Detailed solution should be shown and all final answers should be encircled. Write out
your solution as shown below:
Sample
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.
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
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
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
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
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
LINES AND PLANES IN
SPACE
MATH23
MULTIVARIABLE CALCULUS
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
Lines
VECTOR VALUED
FUNCTIONS
MATH23
MULTIVARIABLE CALCULUS
Definition: Vector-Valued Functions
Let r(t) be a vector-valued function in 3-space, then
for each allowable value of t (called domain) r = r(t)
can be written as
r = r(t) = < f1(t),f2(t),f3(t) >
Examp
IMPROPER INTEGRALS
MATH23
MULTIVARIABLE CALCULUS
GENERAL
OBJECTIVE
At the end of the lesson the students are
expected to:
Identify an Improper Integral
Classify Improper Integral
Evaluate improper integrals by applying limits in the
definite integral
Appl
VECTOR ALGEBRA
MATH23
MULTIVARIABLE CALCULUS
GENERAL
OBJECTIVE
At the end of the lesson the students are
expected to:
Define Vectors and Differentiate it against scalar
quantities
Find the norms of a vector
Perform Basic Algebraic with given vectors
Ve
MULTIVARIABLECALCULUSLONG
QUIZ1
VECTORSIN3SPACE
Vectors in 2space (normal two dimensional space) and 3space (normal three
dimensionalspace).
Avectorcanberepresentedgeometricallybyadirectedlinesegmentthatstartsata
pointA,calledtheinitialpoint,andendsatapoi
Vectors
Vectors and Scalars
Scalar Quantity a quantity that can be
completely described by giving its
magnitude only.
Example: speed, time, energy, temperature
Vector Quantity a quantity that can be
completely described by giving its
magnitude and direct
11.7
Cylindrical and Spherical
Coordinates
The Cylindrical Coordinate System
In a cylindrical coordinate system, a point P in
space is represented by an ordered triple r , , z
1.
r , is a polar representation of the projection
P in the xy-plane.
2. z is
Functions of Several Variables
A. Quadric Surfaces
B. Level Curves and Level
Surfaces
C. Partial Derivatives
QUADRIC SURFACES
The equation
Ax 2 By 2 Cz 2 Dxy Exz Fyz Gx Hy Iz J 0
is called a second-degree equation in x, y,
and z.
The graph of such equatio
Functions of Several Variables
A.
B.
C.
Quadric Surfaces
Level Curves and
Level Surfaces
Partial Derivatives
QUADRIC SURFACES
The equation
Ax 2 By 2 Cz 2 Dxy Exz Fyz Gx Hy Iz J 0
is called a second-degree equation in x, y,
and z.
The graph of such equatio
IMPROPER INTEGRALS
Introduction:
In this topic, the concept of the definite
integral will be extended. It will include
infinite intervals of integration and
integrands that become infinite within the
interval of integration which is called
improper integ
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
SPACE COORDINATES AND
SURFACES
MATH23-1
CALCULUS 3
OBJECTIVES:
At the end of the lesson, the student is expected to be
able to:
Define space coordinates.
Plot points of space coordinates.
Write and sketch the graphs of space coordinate
equations.
Know
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:
MATH23X
EXIT EXAMINATION
4th Quarter SY 20112012
Course
Outcome
Problem 1
Choices
Course
Outcome
Problem 2
Choices
Course
Outcome
Problem 3
Choices
Course
Outcome
Problem 4
Solve linear equations and its applications.
The product of two numbers is 16. If
CORNEAL TOPOGRAPHY:
MEASUREMENT AND
SIGNIFICANCE
NORMAL CORNEAL
DIMENSIONS AND
TOPOGRAPHY
Corneal diameter
HVID: 10 14 mm
ave: 11.7 mm
VVID: ave: 10.6 mm
smaller than HVID of about 0.5 1.0 mm
CORNEAL THICKNESS
Clinical Studies:
0.50 0.65 mm
Gullstrands
FACULTY
COURSE CODE/SECTION
DAY
Week #
Date
April 20/21
Week 1
April 22/23
April 27/28
PHY10 Course Calendar for 4th Term 2014-2015
NOVIDA, FE M.
PHY 10
TIME
ROOM
TOPIC/S
Orientation
VECTORS
Introduction to Vector and Scalar
Quantities
May 4/5
May 11/12
D
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
MAPA INSTITUTE OF TECHNOLOGY
Department of Mathematics
VISION
Mapua shall be among the best universities in the world.
MISSION
a. The Institute shall provide a learning environment in order for its students to acquire the attributes that will
make them gl
INTRODUCTION TO SPACE
COORDINATES
SPACE COORDINATES
and
SURFACES
OBJECTIVES:
At the end of the lesson, the student is expected to be
able to:
define space coordinates.
plot points of space coordinates.
write and sketch the graphs of space coordinate
eq