Basic Mathematics Syllabus

 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.

1. Functions and their graphs

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 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.

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, 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.

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, 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

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 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

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