Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners

Introduction to Differential Calculus: Systematic Studies with Engineering Applications for Beginners

Language: English

Pages: 784

ISBN: 1118117751

Format: PDF / Kindle (mobi) / ePub

Enables readers to apply the fundamentals of differential calculus to solve real-life problems in engineering and the physical sciences

Introduction to Differential Calculus fully engages readers by presenting the fundamental theories and methods of differential calculus and then showcasing how the discussed concepts can be applied to real-world problems in engineering and the physical sciences. With its easy-to-follow style and accessible explanations, the book sets a solid foundation before advancing to specific calculus methods, demonstrating the connections between differential calculus theory and its applications.

The first five chapters introduce underlying concepts such as algebra, geometry, coordinate geometry, and trigonometry. Subsequent chapters present a broad range of theories, methods, and applications in differential calculus, including:

  • Concepts of function, continuity, and derivative

  • Properties of exponential and logarithmic function

  • Inverse trigonometric functions and their properties

  • Derivatives of higher order

  • Methods to find maximum and minimum values of a function

  • Hyperbolic functions and their properties

Readers are equipped with the necessary tools to quickly learn how to understand a broad range of current problems throughout the physical sciences and engineering that can only be solved with calculus. Examples throughout provide practical guidance, and practice problems and exercises allow for further development and fine-tuning of various calculus skills. Introduction to Differential Calculus is an excellent book for upper-undergraduate calculus courses and is also an ideal reference for students and professionals alike who would like to gain a further understanding of the use of calculus to solve problems in a simplified manner.

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of y [¼f(x)] at x ¼ x1 and the Slope of its Graph at x ¼ x1 9.4 A Notation for Increment(s) 9.5 The Problem of Instantaneous Velocity 9.6 Derivative of Simple Algebraic Functions 9.7 Derivatives of Trigonometric Functions 9.8 Derivatives of Exponential and Logarithmic Functions 9.9 Differentiability and Continuity 9.10 Physical Meaning of Derivative 9.11 Some Interesting Observations 9.12 Historical Notes 10 Algebra of Derivatives: Rules for Computing Derivatives of Various Combinations of

length of the domain interval and that of range interval should be the same, but this need not be true. For example, recall that the exponential function y ¼ ex, defines a one-to-one mapping from ( À1, 1) onto (0, 1). The inverse of exponential function y ¼ ex is called logarithmic function (expressed by x ¼ loge y), which is defined from (0, 1) onto ( À1, 1). Remark: This is possible due to the fact that infinite sets can have proper subsets (which are also infinite) such that a one-to-one mapping

defined. Thus, we say that f(x) is defined for x 2 (À1, 2) [ (2, 1). In calculus, we study the behavior of functions on intervals, which are defined using the absolute value of a real number. Hence, we introduce the concept of the absolute value of a real number. 3.10.3 Definition of Absolute Value of Real Number(s) If a is any real number, the absolute value of a, denoted by jaj is a, if a is non-negative, and –a if a is negative. Thus, with symbols we write, & jaj ¼ a if a ! 0 Àa if a < 0

non-negative. Distance = a – b b a Distance = b – a a b Examples: 1. 2. 3. 4. 5. j7 À 3j ¼ j4j ¼ 4 j5 À 12j ¼ jÀ7j ¼ À(À7) ¼ 7 j8 À (À3)j ¼ j8 þ 3j ¼ j11j ¼ 11 jÀ2 À (À7)j ¼ jÀ2 þ 7j ¼ j5j ¼ 5 jÀ9 À (À6)j ¼ jÀ9 þ 6j ¼ jÀ3j ¼ À(À3) ¼ 3 Let us consider equations involving absolute values. Example (6): Solve the equation jxj ¼ 5 If x ! 0, then jxj ¼ x ¼ 5 If x < 0, then jxj ¼ Àx ¼ 5 ) x ¼ À5. Hence the solution set is {5, À5}. Example (7): Solve the equation jx À 8j ¼ 7 If (x À 8) ! 0, then

difficult to grasp the concept (of derivatives) with our systematic approach. The relationship between f(x) and f 0 (x) is the main theme. We will study what it means for f 0 (x) to be “the rate function” of f(x), and what each function says about the other. It is important to understand clearly the meaning of the instantaneous rate of change of f(x) with respect to x. These matters are systematically discussed in this book. Note that we have answered the first two questions and now proceed to

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