# Calculus I

## Spring 2013/2014

### Announcements

Final Exam  will be held on 09.06.2014, 8.30

Topics for Final Exam:

1. Linear Approximation

2. L'Hopital's Rule

3. Antiderivatives

4. Fundamental Theorem of Calculus

5. Working with Integrals

6. Substitution Rule

7. Region between Curves

8. Integration by Parts

9. Trigonometric Integrals

10. Trigonimetric substitution

11. Partial Fractions

12. Improper Integrals

13. Sequences

14. Infinite Series

15. Divergence and Integral Test

16. Ratio, Root and Comparison Test

17. Alternating Series

A cell phone was found in one of the exam class rooms. The person who lost it can come and take it in AS 138.

• ### Topic 1

COURSE OBJECTIVE

Calculus was first invented to meet the mathematical needs of scientists of the sixteenth and seventeenth centuries, needs that mainly mechanical in nature. Nowadays it is a tool used almost everywhere in the modern world to describe change and motion. Its use is widespread in science, engineering, medicine, business, industry, and many other fields. Calculus also provides important tools in understanding functions and has led to the development of new areas of mathematics including real and complex analysis, topology, and non-euclidean geometry.

The objective of this course is to introduce the fundamental ideas of the differential and integral calculus of functions of one variable.

CATALOG DATA

Limits and continuity. Derivatives, Rules of differentiation, Higher order derivatives, Chain rule. Related rates. Rolle's and the mean value theorem. Critical Points. Asymptotes. Curve sketching. Integrals, Fundamental Theorem, Techniques of integraion, Definite integrals. Application to geometry and science. Indeterminate forms. L'Hospital's Rule. Improper integrals. Polar Coordinates.

RELATIONSHIP WITH THE OTHER COURSES

This course is the first of a series of engineering mathematics courses. It is a prerequisite to Math152 - Calculus II, and to Math203 Ordinary Differential Equations (or similar courses on differential equations).

LEARNING OUTCOMES

On succesful completion of the course, the students should be able to:

• recognise properties of functions and their inverses;
• recall and use properties of polynomials, rational functions, exponential, logarithmic, trigonometric and inverse-trigonometric functions;
• understand the terms domain and range;
• sketch graphs, using function, its first derivative, and the second derivative;
• use the algebra of limits, and l’Hôpital’s rule to determine limits of simple expressions;
• apply the procedures of differentiation accurately, including implicit and logarithmic differentiation;
• apply the differentiation procedures to solve related rates and extreme value problems;
• obtain the linear approximations of functions and to approximate the values of functions;
• perform accurately definite and indefinite integration, using parts, substitution, inverse substitution;
• understand and apply the procedures for integrating rational functions;
• perform accurately improper integrals;
• calculate the volumes of solid objects, the length of arcs and the surface area;
• perform polar-to-rectangular and rectangular-to-polar conversions.

# Limits and Continuity

Section 2.1-2.2: The Idea of Limits, Definitions of Limits (1 hours)
Section 2.3: Techniques for Computing Limits (1 hour)
Section 2.4: Infinite Limits (1 hour)
Section 2.5: Limits at Infinity (1 hour)
Section 2.6: Continuity (1 hour)

# Derivatives

Section 3.1: Introducing the Derivative (1 hour)
Section 3.2: Rules of Differentiation (1 hour)
Section 3.3: Product and Quotient Rule (1 hour)
Section 3.4: Derivatives of Trigonometric Functions (1 hour)
Section 3.6: Chain Rule (1 hour)
Section 3.7: Implicit Differentiation (1 hour)
Section 3.8: Derivatives of Logarithmic and Exponential Functions (1hour)
Section 3.9: Derivatives of inverse trigonometric Functions (1 hour)

# Application of the Derivative

Section 4.1: Maxima and Minima (1 hour)
Section 4.2: What Derivatives Tell us (2 hours)
Section 4.3: Graphing Functions (2 hours)
Section 4.5: Linear Approximations (1 hour)
Section 4.6: The Mean Value Theorem (1 hour)
Section 4.7: L’Hopital’s Rule (2 hours)
Section 4.8: Antiderivatives (1 hour)

# Integration

Section 5.2: The Definite Integral (1hour)
Section 5.3: Fundamental Theorem of Calculus (0.5 hour)
Section 5.4: Working with Integrals (1 hour)
Section 5.5: Substitution Rule (1.5 hours)
Section 5.3: Fundamental Theorem of Calculus (0.5 hour)
Section 5.4: Working with Integrals (1 hour)
Section 5.5: Substitution Rule (1.5 hours)

# Application of Integration

Section 6.2: Region between Curves (2 hours)

# Integration Techniques

Section 7.1 Integration by Parts (2 hours)
Section 7.2 Trigonometric Integrals (2 hours)
Section 7.3 Trigonometric Substitution (2 hours)
Section 7.4 Partial Fractions (2 hours)
Section 7.7 Improper Integrals (2 hours)

# Sequences and Infinite Series

Section 8.1-8.2: Sequences (2 hours)
Section 8.3: Infinite Series (2 hours)
Section 8.4: Divergence and Integral Test (1 hour)
Section 8.5: Ratio, Root, and Comparison Test (1,5 hours)
Section 8.6: Alternating Series (1,5 hours)