Caraga Regional Science High School San Juan Surigao City

MATHEMATICS IV Syllabus-Calculus

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.

Course Objectives

After undergoing the fourth year course in mathematics, the learner shall be able to:

Define intuitively and understand the concepts in differential and basic Integral Calculus

Use Differential Calculus in graphing

Apply the concepts of Differential in solving problems on optimization, related rates.

Use definite integrals to evaluate area of a region under a curve

Graphically interpret derivatives of functions

Communicate with clarity using mathematics as language.

Gain confidence in problem solving through reasoning and application of concepts in Differential and basic Integral Calculus.

Appreciate the interrelationship of Differential and basic Integral Calculus and their applications to other discipline

Credit Unit: 1.5

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.

Equations of Lines

Equations of Circles

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

Function and their Graphs

LIMITS AND CONTINUITY OF ALGEBRAIC FUNCTIONS

Specific Objectives

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.

Topics: The Limit of a Function, Theorems on Limits of

Functions, One-sided Limits, Infinite Limits, and Limits at Infinity SECOND QUARTER

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

THE DERIVATIVE AND DIFFERENTIATION

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

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

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

MAXIMA AND MINIMA INVOLVING ALGEBRAIC FUNCTIONS

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.

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

INTEGRATION

Specific Objectives

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

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

San Juan Surigao City

MATHEMATICS IVSyllabus-Calculus## 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.

After undergoing the fourth year course in mathematics, the learner shall be able to:Course ObjectivesCredit Unit: 1.5Time Duration: 60 min/dayMission: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 developmentFourth YearGeneral and Specific CompetenciesAfter undergoing the fourth year course in Mathematics IV, the learner shall have developed the following competencies:

FIRST QUARTER

I.PRE- REQUISITE TOPICSThese 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.

- Equations of Lines
- Equations of Circles
- Inequalities

3.1 Intervals3.1.1Open 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

- Function and their Graphs
- Specific Objectives
- Demonstrate understanding and manifest skills in finding limits and continuity of algebraic functions.

1.1 Explain the concept of a limit intuitively byLIMITS AND CONTINUITY OF ALGEBRAIC FUNCTIONSgraph 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.

Functions, One-sided Limits, Infinite Limits, andTopics:The Limit of a Function, Theorems on Limits ofLimits at Infinity

SECOND QUARTER

1.15 Define continuity of a function at a pointSpecific Objectives(continuation)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

- Manifest ability and skills in finding the derivative of algebraic functions.

2.1 Define intuitively what a tangent line to a function isTHE DERIVATIVE AND DIFFERENTIATION2.2 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

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 RuleTopics:THIRD QUARTER

2.13 Illustrate the derivative as the velocity of anSpecific Objectives (Continuation)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

- 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 functionMAXIMA AND MINIMA INVOLVING ALGEBRAIC FUNCTIONS3.2 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.

: 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 ExtremaTopicsFOURTH QUARTER- INTEGRATION
- Demonstrate understanding of the integral.

4.1Define the anti-derivative of a functionSpecific Objectives4.2State 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

Antidifferentiation, Some Techniques of Antidifferentiation, Area, The Definite Integral, Properties of the Definite IntegralTopics:REFERENCES:The Calculus with Analytic Geometry, 6th Ed., by Louis LeitholdLesson plans