Single Variable Calculus

Yunzhi Zou, Beijing World Publishing Corporation

• Publisher: De Gruyter
• Year: 2018
• ISBN:
  9783110527858
• Edition: 1
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

The book is a comprehensive yet compressed entry-level introduction on single variable calculus, focusing on the concepts and applications of limits, continuity, derivative, defi nite integral, series, sequences and approximations. Chapters are arranged to outline the essence of each topic and to address learning diffi culties, making it suitable for students and lecturers in mathematics, physics and engineering.

Contents
Prerequisites for calculus
Limits and continuity
The derivative
Applications of the derivative
The definite integral
Techniques for integration and improper integrals
Applications of the definite integral
Infinite series, sequences, and approximations

1Prerequisites for calculus

1.1Overview of calculus

In this chapter, you will:

–see a big picture of calculus;

–review the definition of a function;

–review properties of a function;

–review basic functions and form new functions using basic functions.

When first introduced to calculus, one probably will ask questions such as “what is calculus?” and “what can calculus do?”. This book has answers to these questions. To begin, we start with two problems, the area problem and the tangent problem.

The area problem

We start with an example. Many students start their school day by traveling in cars or buses from their homes to their schools. We will simplify our discussion by assuming such a trip is along a straight road of 100 km and the trip duration is 2 hours. If you kept looking at the speedometer (see Figure 1.1.1), then you would be able to sketch an approximate speed/velocity versus time graph.

The velocity graph in Figure 1.1.2 shows a period of acceleration, followed by a brief period of constant velocity and then a period of deceleration.

Figure 1.1.1: A speedometer.

Figure 1.1.2: Velocity versus time graph.

How do we know the distance traveled up to time = 1 hour? We note that

$\text{distance}\text{traveled}=\text{speed}\times \text{time}\text{.}$

However, the problem here is that the speed/velocity on this trip is not constant, and in order to determine the distance traveled, we subdivide the time interva...