Download - Chapter 1 (Preliminaries)
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Applied Calculus &
Analytical Geometry
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About the course!
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Books to follow!
• Course Book– CALCULUS by Thomas 11th Edition
(Compulsory)
• Reference Book– Calculus by Howard Anton 7th edition
(Optional)
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Grading Policy
• Assignment 20%
• Quizzes 10%
• Midterm Exam 20%
• Final Exam 50%
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Requirements
• Course Text book (Thomas Calculus 11th ed)• Separate Register only for Calculus I
(Because only those interested student’s questions will be entertained who will work continuously!!!)
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What is Calculus?
• Calculus is the mathematics of Change.• Applied when we have
– Motion– Growth– Variable forces producing acceleration
• Invented to meet mathematical needs of scientists of 16th and 17th centuries.– The science of that time was mechanical in
nature
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What is Calculus?
• The key elements were put in place, independently, by Newton (1642-1727) and Leibniz (1646-1716).
• The range of applications of calculus to ‘real world problems’ is vast and growing all the time.
• Two types– Differential calculus– Integral calculus
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Differential Calculus
• Differential calculus deals with problems of “rates of change”– Finding slopes of curves– Velocities and accelerations of moving bodies– Find firing angles to have greatest range– Predict time when planets would be closest
together or farther apart
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Integral Calculus
• Integral calculus deals with “determining a function from information about its rate of change”– Future location of a moving body– Areas of irregular regions in the plane– Measure lengths of curves– Find volumes and masses of arbitrary solids
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Learning Calculus
• Read the text (Most Important)• Do the assignments as follows
– Sketch diagrams wherever possible, it will help in enhancing your visualization ability
– Write your solutions in connected step by step logical fashion, as if you are explaining to someone else
– Think about why each exercise is there? Why was it assigned? How is it related to the other assigned exercises?
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Last words before start
• Please be in time (Don’t come late!)
• Keep your mobiles off (or silent) during class
• Don’t miss any class
• Solve assignments yourself !
• Handover assignments on time to get full grade!
• Don’t miss any quiz
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Let us Start !!!
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Chapter 1
Preliminaries
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Topics to be covered
• Real Numbers and Real Line
• Lines, Circles and Parabolas
• Functions and their Graphs
• Identifying Functions• Combining functions, shifting and scaling graphs
• Trigonometric Functions
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1.1-Real Numbers and the Real Line
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The Real Line
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Rules for Inequality
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Main Subsets of Real Line
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Rational and Irrational Numbers
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Irrational Numbers
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Interval
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Example (Solving Inequality)
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Absolute Value Function
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Properties of Abs( )
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Example
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Exercise 1.1Practice Exercise (5—34 all odd problems)
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Announcement
Please bring Low Priced Text book
• Thomas calculus (11th edition) and
• Separate Register for Calculus with you
On next
With your name written on it!
Every such student will be given 10 marks as part of (quiz)!!!!
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Cartesian Coordinates in the Plane
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Signs in Quadrants
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Increments
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Slope
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Equation of Straight Line (Point Slope form)
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Example
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A Line through two points
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Slope Intercept form
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Example
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Parallel and Perpendicular Lines
• Two lines are parallel if they have same slope. Conversely, if two lines have same slope then they are parallel.
• Two lines are perpendicular if product of their slopes is -1.That is,
m1 * m2 = -1
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Distance formula in plane
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Example
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Equation of Circle
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Example
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Finding a circle’s radius and center
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Interior and Exterior
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Parabolas
Parabolas arise as graphs of:
ax2+bx+c = 0
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Graph of eq y=ax2
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Graph of ax2+bx+c
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Role of a
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Example
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Exercise 1.2
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Assignment #
Dead Line()
Ex 1.1 ( 7,9,11,15,17,25,27,31,33)
Ex 1.2 ( 1 to 45 and 53 to 60 odd)
Write down your name, subject, Section, ID, Assignment #
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Function and their Graphs
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Mapping
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Some examples
f(x)= x+5
f(x)=x2+5x
f(x)= Sin(x)
f(x)=cos(x)
f(x)=5+sin(x)
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Domain and Range
y = f(x)
DomainSet from where function (f) accepts values
Range (Co domain or Image set)
Set on which function (f) maps those values
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Always remember that we shall consider Real values only not complex values throughout this course (Cal 1 and Cal 2)
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Example
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Example
1.
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Greatest Integer Function
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Least Integer Function
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Writing formula for piecewise defined functions
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Exercise 1.3
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Identifying Function and Mathematical Models
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Types of Functions
• Linear functions• Power functions• Polynomials• Algebraic functions• Trigonometric functions• Exponential functions• Logarithmic function• Transcendental functions
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Linear functions
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Power functions
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Case 1 (xa ,a=n, a positive integer)
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Case 2 (xa ,a=-1 or a=-2)
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Case 3 (xa ,a=1/2,1/3,2/3,3/2)
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Case 4 (Polynomial)
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Case 5 (Rational Functions)
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Summary
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Summary
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Graphs of Algebraic Functions
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Graph of Sine and Cosine
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Graph of Exponential Functions
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Graph of Logarithmic Functions
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Increasing and Decreasing functions
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Even and Odd functions
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Examples of Even and Odd functions
Even
Odd
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Proportionality
Example
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Mathematical Model
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Example of proportionality
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Exercise 1.4(1--30)
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Combining Functions; shifting and Scaling
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Example
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Composite Functions
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Example
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Shifting Graph
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Example
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Scaling and Reflecting Graph
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Example
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Example
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Ellipses
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Exercise 1.5(1--50)
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Trigonometric Functions
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Conversion from Radian to Degree and from Degree to Radian
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Example
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Angle Convention
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Important Quotients
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Important Table to Cram !!!
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Periodic Functions
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Another to Cram (Also for Limits) !
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Periods of Trigonometric Functions
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Even and Odd
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Identities
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Addition Formulas
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Double Angle Formulas (Imp)
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Half Angle Formula (Imp)
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The Law of Cosine
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End of Chapter # 1