Caraga Regional Science High School
San Juan Surigao City


MATHEMATICS IV
Syllabus-Calculus


  1. Course Description


Mathematics IV is a Course given to fourth year students of the Regional Science High School. The course covers both Differential and Basic Integral Calculus that includes the study of Limits, Differentiation, Basic Integration, and Areas under Curves.
This course provides the regional Science High School student with a distinct advantage over ordinary high school in terms of preparation for college mathematics, especially if the student is to take up a science or engineering course.
  1. Course Objectives
After undergoing the fourth year course in mathematics, the learner shall be able to:
  1. Define intuitively and understand the concepts in differential and basic Integral Calculus
  2. Use Differential Calculus in graphing
  3. Apply the concepts of Differential in solving problems on optimization, related rates.
  4. Use definite integrals to evaluate area of a region under a curve
  5. Graphically interpret derivatives of functions
  6. Communicate with clarity using mathematics as language.
  7. Gain confidence in problem solving through reasoning and application of concepts in Differential and basic Integral Calculus.
  8. Appreciate the interrelationship of Differential and basic Integral Calculus and their applications to other discipline
  9. Credit Unit: 1.5
  10. Time Duration: 60 min/day



Mission: Towards its goal, the Caraga Regional Science High School is committed to provide quality education that is equitably accessible to the intellectuality gifted and science inclined youth who understand and internalize the value of scientific knowledge towards the advancement of our country.




Vision:The Caraga Regional Science High School aims to develop a core of Youth who are scientifically inclined science oriented and competent whose scientific efforts shall lead the country to progress and development









Fourth Year

General and Specific Competencies


After undergoing the fourth year course in Mathematics IV, the learner shall have developed the following competencies:
FIRST QUARTER
I. PRE- REQUISITE TOPICS
These topics are to be taken up in the first four weeks as part of the necessary preparation needed by a student going into calculus. The teacher must ensure that the students are proficient in these topics before beginning calculus.
  1. Equations of Lines
  2. Equations of Circles
  3. Inequalities
3.1 Intervals
3.1.1Open intervals
3.1.2Closed intervals
3.1.3Half – open intervals
3.2 Linear Inequalities
3.3 Quadratic Inequalities
3.4 Polynomial Inequalities
3.5 Rational Inequalities
3.6 Absolute Value of Inequalities
  1. Function and their Graphs
  2. LIMITS AND CONTINUITY OF ALGEBRAIC FUNCTIONS
    1. Specific Objectives
    2. Demonstrate understanding and manifest skills in finding limits and continuity of algebraic functions.
1.1 Explain the concept of a limit intuitively by
graph and by table of values
1.2 State the properties of limits
1.3 Find limits of algebraic functions
1.3.1 Linear
1.3.2 Quadratic
1.3.3 Higher degree polynomials
1.3.4 Rational
1.3.5 Functions involving radicals
1.4 Identify the different indeterminate forms
(specifically 0/0 and infinity over infinity) and
algebraically manipulate the functions whose
limits lead to these in order to fin limits.
1.5 Illustrate the concept of one sided limits
through graphs
1.6 Evaluate on-sided limit
1.7 Determine if the limit of a function exists by
using the concept of one sided limits.
1.7.1 Explain the concept of infinite limits intuitively by graph and by table of values
1.8 State the properties of infinite limits
1.9 If the limit of a function does not exist,
identify when this limit positive or negative
infinity
1.10 State the relationship between an infinite limit
and a vertical asymptote
1.11 Explain the concept of limits at infinity
intuitively by graph and table
1.12 State the properties of limits at infinity
1.13 Find the limits of function f(x) as x approaches
positive or negative infinity
1.14 State the relationship between a limit at
infinity and a horizontal asymptote.
  1. Topics: The Limit of a Function, Theorems on Limits of
Functions, One-sided Limits, Infinite Limits, and
Limits at Infinity
SECOND QUARTER
  1. Specific Objectives(continuation)
1.15 Define continuity of a function at a point
1.16 Use the definition of continuity to determine
whether a function is continuous or not at a
given point
1.17 Identify the possible points of discontinuity of
a function
1.18 Sketch the graph of continuous and discontinuous
function
1.19 Define continuity within an interval
1.20 Identify whether a function is continuous or not
within a given interval
  1. THE DERIVATIVE AND DIFFERENTIATION
    1. Manifest ability and skills in finding the derivative of algebraic functions.
2.1 Define intuitively what a tangent line to a function is
2.2 Use the concept of limits to find the slope of a tangent line to a function
2.3 Find the slope of the tangent line to a function at a given point
2.4 Find the equation of the tangent line to function at a given point
2.5 Define a normal line to a graph of a function at a given point
2.6 Find the equation of a normal line
2.7 Define the derivative of a function
2.8 Find the derivative of a function using the definition
2.9 State the rules on differentiation
2.10 Apply the rules for finding the derivative of algebraic functions
2.11 State the chain rule
2.12 Find the derivative of composite function using the
chain rule
  1. Topics:
Continuity of a Function at a Number, Continuity of a Composite Function and Continuity on an Interval, Tangent Line and the Derivative, Differentiability and Continuity , Theorems on Differentiation of Algebraic Functions, The Derivative of a Composite Function and the Chain Rule
THIRD QUARTER
  1. Specific Objectives (Continuation)
2.13 Illustrate the derivative as the velocity of an
object in rectilinear motion
2.14 Illustrate the derivative as a rate of change
2.15 Solve problems on rates of change
2.16 Find the second, third and other higher-ordered
derivatives of an algebraic function
2.17 Differentiate implicit functions
2.18 Solve related rates problems using implicit
differentiation
  1. MAXIMA AND MINIMA INVOLVING ALGEBRAIC FUNCTIONS
    1. Demonstrate understanding and manifest skills in solving word problems on optimization and related rates involving algebraic functions
3.1 Define a relative maximum/minimum value of a function
3.2 Identify intervals where function is increasing or
decreasing
3.3 Determine whether a function is increasing or
decreasing in given intervals using the derivative
3.4 Find the critical values of a function.
3.5 Apply the first derivative test to determine if a
critical point is a relative maximum or a relative
minimum of an algebraic function
3.6 Determine when a function is concave up or concave
down
3.7 Identify the relationship between the second
derivative and concavity
3.8 Define a point of inflection
3.9 Identify the relationship between the second
derivative and a point of inflection
3.10 Use the second derivative test in finding the
relative extrema of an algebraic function
3.11 State the extreme value theorem.
3.12 Find absolute extrema of a function in a given closed
interval.
3.13 Solve maximization/minimization word problems.
  1. Topics: Rectilinear Motion and the Derivative as a rate of Change, Related Rates, Derivatives of Higher Order, Maximum and minimum functions values, Increasing and Decreasing Functions and the First Derivative test, Concavity and Points of Inflection, The Second Derivative Test for relative Extrema

FOURTH QUARTER
  1. INTEGRATION
    1. Specific Objectives
    2. Demonstrate understanding of the integral.
4.1Define the anti-derivative of a function
4.2State the properties of anti-differentiation
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
4.3Define definite integral using area
4.4State the fundamental theorem
4.5Evaluate definite integral using fundamental theorems of calculus
  1. Topics: Antidifferentiation, Some Techniques of Antidifferentiation, Area, The Definite Integral, Properties of the Definite Integral

REFERENCES:The Calculus with Analytic Geometry, 6th Ed., by Louis Leithold
Lesson plans