This page contains Syllabus of Basic Mathematics of BIT.

Title | Basic Mathematics |

Short Name | |

Course code | MTH104 |

Nature of course | Theory |

Semester | first-semester |

Full marks | 80+20 |

Pass marks | 32 + 8 |

Credit Hrs | 3 |

Elective/Compulsary | Compulsary |

### Course Description

** Course Description:**

This course familiarizes students with functions, limits, continuity, differentiation, integra-

tion of function of one variable, logarithmic, exponential, applications of derivative and

antiderivatives, differential equations, partial derivatives.

**Course Objectives:**

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.

### Units and Unit Content

- 1. Functions Limits and Continuity
- teaching hours: 5 hrs
Definition, domain range, Graphs of functions, Representing a function numerically, the vertical 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 graphsSums, differences, products and quotients, Composite functions, Shifting a graph of a function.

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):

1.5: Exponential functions: Definition, Exponential behavior, Exponential growth and decay.

Worked out examples: 1-4.

**Exercises 1.5**: 29-331.6: Inverse Functions and Logarithms

Worked out examples: 1 - 4, 6, 7.

**Exercises 1.5**: 79 - 812.1: Rate of change and tangent to curves.

Worked out examples: 1-5.

**Exercises 2.1**: 1, 3, 6, 7, 9, 15, 17.- 2. Limits and Continuity
- teaching hours: 3 hrs
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

- 3. Differentiation
- teaching hours: 5 hrs
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, 253.2 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, logarithm, exponential functions and ideas of chain rules.

3.10 Related rates

Worked out Examples: 1-6.

- 4. Application of Differentiation
- teaching hours: 5 hrs
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

- 5. Integration
- teaching hours: 5 hrs
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- 6. Applications of Definite Integrals
- teaching hours: 3 hrs
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

- 8. Techniques of Integrations
- teaching hours: 5 hrs
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

- 8. First Order Differential Equations
- teaching hours: 4 hrs
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, 139.2 First order linear equation

Worked out examples 1, 2, 3

**Exercises 9.2**: 1-10, 15-219.3 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

- 9. Infinite Sequence and Series
- teaching hours: 5 hrs
10.1. Sequences

Worked out Examples: 1-11

**Exercises 10.1**: 1,2,3,7,8.13,16,27-32 Infinite seriesWorked 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- 10. Partial Derivatives
- teaching hours: 5 hrs
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

### Lab and Practical works