# Basic Mathematics - Syllabus

Embark on a profound academic exploration as you delve into the Basic Mathematics course () within the distinguished Tribhuvan university's BIT department. Aligned with the BIT Curriculum, this course (MTH104) seamlessly merges theoretical frameworks with practical sessions, ensuring a comprehensive understanding of the subject. Rigorous assessment based on a 80+20 marks system, coupled with a challenging passing threshold of , propels students to strive for excellence, fostering a deeper grasp of the course content.

This 3 credit-hour journey unfolds as a holistic learning experience, bridging theory and application. Beyond theoretical comprehension, students actively engage in practical sessions, acquiring valuable skills for real-world scenarios. Immerse yourself in this well-structured course, where each element, from the course description to interactive sessions, is meticulously crafted to shape a well-rounded and insightful academic experience.

** 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

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 graphs

Sums, 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**: 29-33

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.

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

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, logarithm, exponential functions and ideas of chain rules.

3.10 Related rates

Worked out Examples: 1-6.

Application of Differentiation

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

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

Applications 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

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

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

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

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

#### Lab works