For First Year Bsc. Mathematics Students

### Calculus of Single Variable - 1

Module ­I
(Functions and Limits)

0.2: Functions and their Graphs­ Definition of a Function, Describing Functions, Evaluating Functions, Finding the Domain of a Function, The Vertical Line Test, Piecewise Defined Functions, Even and Odd Functions (quick review).
0.4: Combining functions­ Arithmetic Operations on Functions, Composition of Functions, Graphs of Transformed Functions , Vertical Translations, Horizontal Translations, Vertical Stretching and Compressing, Horizontal Stretching and Compressing, Reflecting.
1.1: Intuitive introduction to Limits­ A Real­ Life Example, Intuitive Definition of a Limit, One­ Sided Limits, Using Graphing Utilities to Evaluate Limits.
1.2: Techniques for finding Limits­ Computing Limits Using the Laws of Limits, Limits of Polynomial and Rational Functions, Limits of Trigonometric Functions, The Squeeze Theorem.
1.3: Precise Definition of a Limit­ ε − δ definition,A Geometric Interpretation, Some illustrative examples.
1.4: Continuous Functions­ Continuity at a Number, Continuity at an Endpoint, Continuity on an Interval, Continuity of Composite Functions, Intermediate Value Theorem.
1.5: Tangent Lines and Rate of change­ An Intuitive Look, Estimating the Rate of Change of a Function from Its Graph, More Examples Involving Rates of Change, Defining a Tangent Line, Tangent Lines, Secant Lines, and Rates of Change.
2.1: The Derivatives­ Definition, Using the Derivative to Describe the Motion of the Maglev, Differentiation, Using the Graph of f to Sketch the Graph of f′ Differentiability, Differentiability and Continuity.
.4: The role of derivative in the real world­ Motion Along a Line, Marginal Functions in Economics.
2.9: Differentials and Linear Approximations­ increments, Differentials, Error Estimates, Linear Approximations, Error in Approximating ∆y by dy.
Module­ II
(Applications of the Derivative)

3.1: Extrema of Functions ­Absolute Extrema of Functions, Relative Extrema of Functions , Fermat’s Theorem , Finding the Extreme Values of a Continuous Function on a Closed Interval, An Optimization Problem.
3.2: The Mean Value Theorem­ Rolle’s Theorem, The Mean Value Theorem, Some Consequences of the Mean Value Theorem, Determining the Number of Zeros of a Function.
3.3: Increasing and Decreasing Functions­ definition , inferring the behaviour of function from sign of derivative, Finding the Relative Extrema of a Function, first derivative test.
3.4: Concavity and Inflection points­ Concavity, Inflection Points, The Second Derivative Test, The Roles of f ′ and f ′′ in Determining the Shape of a Graph.
3.5: Limits involving Infinity; Asymptotes­ Infinite Limits, Vertical Asymptotes, Limits at Infinity, Horizontal Asymptotes, Infinite Limits at Infinity, Precise Definitions.
3.6: Curve Sketching ­The Graph of a Function, Guide to Curve Sketching, Slant Asymptotes , Finding Relative Extrema Using a Graphing Utility.
3.7: Optimization Problems – guidelines for finding absolute extrema , Formulating Optimization Problems ­ application involving several real life problems.
Module ­III
(Integration)
4.1: Anti derivatives, Indefinite integrals, Basic Rules of Integration, a few basic integration formulas and rules of integration, Differential Equations, Initial Value Problems.
4.3: Area­ An Intuitive Look, The Area Problem, Defining the Area of the Region Under the Graph of a Function­ technique of approximation, An Intuitive Look at Area (Continued), Defining the Area of the Region Under the Graph of a
Function­ precise definition , Area and Distance.
4.4: The Definite Integral­ Definition of the Definite Integral, Geometric Interpretation of the Definite Integral, The Definite Integral and Displacement, Properties of the Definite Integral , More General Definition of the Definite Integral.
4.5: The Fundamental Theorem of Calculus­ How Are Differentiation and Integration Related?, The Mean Value Theorem for Definite Integrals, The Fundamental Theorem of Calculus: Part I, inverse relationship between differentiation and integration, Fundamental Theorem of Calculus: Part 2, Evaluating Definite Integrals Using Substitution, Definite Integrals of Odd and Even Functions, The Definite Integral as a Measure of Net Change
Module ­IV
( Applications of Definite Integral )

5.1: Areas between Curves­ A Real Life Interpretation, The Area Between Two Curves, Integrating with Respect to y­ adapting to the shape of the region, What Happens When the Curves Intertwine?
5.2: Volume – Solids of revolution, Volume by Disk Method, Region revolved about the x­ axis, Region revolved about the y­axis , Volume by the Method of Cross Sections .
5.4: Arc Length and Areas of surfaces of revolution­ Definition of Arc Length, Length of a Smooth Curve, arc length formula , The Arc Length Function , arc length differentials , Surfaces of Revolution, surface area as surface of revolution.
5.5: Work­ Done by a Constant Force, Work Done by a Variable Force, Hook’s Law, Moving non rigid matter, Work done by an expanding gas.
5.7: Moments and Center of Mass­ Measures of Mass, Center of Mass of a System on a Line, Center of Mass of a System in the Plane, Center of Mass of Laminas.