BIT Basic Mathematics

This course familiarizes students with functions, limits, continuity, differentiation, integration of function of one variable, logarithmic, exponential, applications of derivative andantiderivatives, differential equations, partial derivatives.

1. Students will be able to understand and formulate real world problems into mathe- matical statements. 2. Students will be able to develop solutions to mathematical problems at the level ap- propriate to the course. 3. Students will be able to describe or demonstrate mathematical solutions either numer- ically or graphically.

Unit 1 Functions and their graphs 

Definition, domain range, Graphs of functions, Representing a function numerically, the ver-
tical line test for a function, Piecewise defined functions, Increasing and decreasing functions,

Even and odd function, Common functions: linear, power, polynomial, rational functions
All worked out examples of 1.1.
Exercises 1.1: 1-8, 15, 18, 23, 25, 26.
1.2: Combining functions:Shifting and Scaling graphs

Sums, differences, products and quotients, Composite functions, Shifting a graph of a func-
tion.

Worked out examples: 1-5
Exercises 1.2: 1-8.
1.4: Graphing with calculator and computers (desmos may be easy) to plot the graph of the
functions (some of the functions):
y = x, y = x^2 , y =1/1 − x, y = sin x, y = cos x, y = sin 100x

1.5: Exponential functions: Definition, Exponential behavior, Exponential growth and decay.
Worked out examples: 1-4.
Exercises 1.5: 29-33
1.6: Inverse Functions and Logarithms
Worked out examples: 1 - 4, 6, 7.
Exercises 1.5: 79 - 81
2.1: Rate of change and tangent to curves.
Worked out examples: 1-5.
Exercises 2.1: 1, 3, 6, 7, 9, 15, 17.

Unit 2. Limits and continuity

2.2 Limit of a Function and Limit Laws
Limits of function values, The limit laws, Eliminating zero denominators algebraically, The
Sandwich theorem(no proof).
Worked out examples: 1-11
2.3 The Precise Definition of a Limit

Definition of limit
Worked out examples: 1-5
One sided limit: Worked out Examples 1-4
2.5 Continuity
Worked out examples: 2, 3
Intermediate Value Theorem for Continuous Functions
Worked out examples: 11, 12
2.6 Limits Involving Infinity; Asymptotes of Graphs
Worked out examples 1, 2, 3
Horizontal Asymptotes
Worked out examples: 4-9
Oblique asymptotes
Worked out examples: 10-14
Vertical asymptotes
Worked out examples: 15-19.
Some related problems

Unit 3. Differentiation

3.1 Tangents and the Derivative at a Point
Finding a Tangent to the Graph of a Function
Rates of Change: Derivative at a point
Worked out Examples: 1, 2
Exercises 3.1: 5-8, 11, 12, 13, 23, 24, 25
3.2 The Derivative as a function
Worked out Examples: 4, 5
Differentiable Functions are continuous
3.4 The Derivative as a rate of change

Worked out Examples: 1-7 Ideas of derivatives of trigonometric, inverse trigonometric, log-
arithm, exponential functions and ideas of chain rules.

3.10 Related rates
Worked out Examples: 1-6.

Unit 4 Application of Derivative 

4.1 Extreme values of functions: Introduction
Worked out examples: 1-4
Exercise 4.1: 21, 22, 23, 31, 32
4.2 The mean value theorem
Rolle’s Theorem(no proof), Lagrange mean value theorem(no proof)
Worked out examples: 1-4
4.3 Monotonic functions and the first derivative test
Increasing functions and decreasing Functions
Worked out examples: 1, 2, 3
4.4 Concavity and curve sketching
Worked out examples: 1-9
4.5 Indeterminate Forms and LHpitals Rule
Indeterminate form, LHpitals rule
All worked out examples
Exercises 4.5: 1-7, 13, 15. 4.6 Applied optimization
Worked out examples: 1-5
4.7 Newton’s method.
Worked out examples: 1, 2
Examples 4.7: 1-4

Unit 5 Integration 

4.8 Antiderivatives
Worked out examples: 1, 2, 3
5.1 Area and Estimating with Finite Sums
Area
worked out examples: 1-4
Exercises 5.1: 1-4
5.2 Sigma notation and limits of finite sums
Worked out examples: 1-5
5.3 The definite integral
Worked out example: 4, 5
5.4 The fundamental theorem of calculus
Mean value theorem for definite integrals, Fundamental theorem of calculus Part 1 and 2
(no proof), The net change theorem
Worked out examples: 2-7
5.5 Indefinite integral and substitution method:
All worked out examples
5.6 Area between the curves
Worked out examples: 4, 5, 6, 7
Exercises 5.6 : 63-66

Unit 6 Application of Definite Integrals 

6.1 Volumes using cylindrical shells
Worked out examples: 1-10
6.2 Volumes using cross-sections
Worked out examples: 2, 3
6.3 Arc length
Worked out examples: 1, 2 3, 4, 5
6.4 Areas of surfaces of revolution
Worked out examples: 1, 2

Unit 7 Techniques of Integrations

Review of integration by parts, trigonometric substitutions, integration of rational functions
by partial fractions. Computer algebra system (Maple)
8.6 Numerical Integration
Numerical Integration
Simpsons Rule: Approximations Using Parabolas
Error Analysis
Worked out examples:1-6
Exercises 8.6: 1, 2, 3, 4, 7, 8, 9, 10. 11, 12, 13, 17, 19, 21.
8.7 Improper integrals
Worked out examples: 1-9

Unit 8 First Order Differential Equations

9.1 Solutions, Slope Fields, and Eulers Method
General first order differential equations and solutions
Worked out examples: 1, 2.
Slope Fields: Viewing Solution Curves
Eulers Method
Worked out examples: 3, 4
Exercises 9.1: 11, 12, 13
9.2 First order linear equation
Worked out examples 1, 2, 3
Exercises 9.2: 1-10, 15-21
9.3 Applications

Motion with resistance proportional to velocity
7.2 Exponential change.
Worked out Examples: 1, 2, 3, 4, 5.
9.4 Graphical solutions of autonomous equations
Example worked out: 1

Unit 9. Infinite Sequence and Series

10.1. Sequences
Worked out Examples: 1-11
Exercises 10.1: 1,2,3,7,8.13,16,27-32 Infinite series
Worked out examples: 1-10
Related problems from exercise 10.2
Ideas of Integral test, comparison test: worked out examples, Alternating series, absolute
and conditional convergence, with at least one worked out examples.
!0.7. Power series
Worked out examples 1-6.
10.8. Taylor and Maclaurin series
Exercises 10. 8: 1, 2 ,3, 4, 7, 9, 11, 12

Unit 10 Partial Derivatives

14.1. Functions of several variables
Worked out examples: 1, 2, 3, 4
14.2 Limits and continuity in higher dimensions
Worked out Examples: 1-6
Exercises: 1, 2, 3, 4, 5, 6, 13, 14
14.3. Partial derivatives
Worked out examples: 1-10
Examples 14.3: 1-18
14.5. Chain rule
Worked out examples: 1-6
14.5. Directional derivative
Worked out examples: 1-5
14.6. Tangent planes and differentials
Worked out examples: 1-4
14.7. Extreme values and saddle points
Worked out examples: 1-5
Exercises 14.7: 1-7