# Read e-book online Calculus Fundamentals Explained PDF

By Samuel Horelick

This textbook is written for everybody who has skilled demanding situations studying Calculus. This publication fairly teaches you, is helping and grasp Calculus via transparent and significant factors of all of the rules, ideas, difficulties and approaches of Calculus, powerful challenge fixing talents and techniques, absolutely labored issues of whole, step by step factors.

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**Example text**

Example: The function f(x) = (x + 1)/x is neither even nor odd : f(– x) = ((_ x) + 1)/(_ x) = 1 _ 1/x = (x _ 1)/x which is not equal to f(x) and is not equal to – f(x) either. CHAPTER 2 – limits LIMITS AND CONTINUITY: A very important concept in Calculus is that of the limit of a function: if f(x) is a function of x, what happens to the value of the function y as x approaches some number? For example, if f(x) = x2 + 3, what is the value of y = f(x) as x approaches 5? 9001 As x approaches 5, the value of y = f(x) approaches 52 + 3 = 28.

Interchanging x and y, we obtain x = y2. Solving for y, we obtain two solutions: positive and negative square roots of x. That is, y = + Öx or y = – Öx, for all x ³ 0. Since for every x there are two different y, this is not a function. Therefore, there is no inverse function for h(x) = x2. Exercises: Find the inverse function f –1(x) if it exists for the following functions: 1. f(x) = 3x 2. f(x) = 4x + 1 3. f(x) = 5 + 2x 4. f(x) = 4(x + 1) 5. f(x) = (1/2)(6x – 3) 6. f(x) = (3x + 5) ¸ 7 7. f(x) = (1/2)(x – 2) + 2 8.

6. Trigonometric functions: sin x, cos x, tan x, cot x, sec x, csc x. SYMMETRY, EVEN AND ODD FUNCTIONS: A function f(x) is called an even function if exchanging – x for x does not change the function. That is, f(– x) = f(x) for all x in the domain of the function. A function f(x) is called an odd function if exchanging – x for x changes the sign of the function. That is, f(– x) = – f(x) for all x in the domain of the function. Examples: If f(x) = x2, then f(– x) = (– x)2 which is the same as f(x) = x2.

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