By Dale W. Johnson
Calculus can attempt the boundaries of even the main complex math scholars. This visible, easy-to-follow publication deconstructs complicated mathematical suggestions in a manner that’s infinitely more uncomplicated to understand. With transparent, color-coded tools, you’ll get step by step directions on fixing difficulties utilizing limits, derivatives, differentiation, curve sketching, and integrals. easy accessibility to options capacity you don’t need to type via long tutorial textual content, and you may seek advice from the Appendix for a glance at universal differentiation ideas, integration formulation, and trigonometric identities.
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Extra resources for Teach Yourself VISUALLY: Calculus
1 chapter Since f(1) < 0 and f(3) > 0, there must be at least one number c in [–2, 12] for which f(c)=0. 3 Set the original function equal to 0. 0 = x3 – x2 – 2x 4 Factor the right-hand side. 0 = x(x2 – x – 2) 0 = x(x – 2)(x + 1) 5 Solve for x. x = 0, x = 2, x = –1 The only value of x the interval (1,3) is 2. Therefore, c = 2. qxp 4/1/08 3:22 PM Page 34 The Intermediate Value and Extreme Value Theorems (continued) Extreme Value Theorem If a function f is continuous on [a,b], then f has both a maximum value and a minimum value on [a,b].
38 Dealing with Indeterminate Forms . . 39 Limits at Infinity: Horizontal Asymptotes . . . . . . . . . . . qxp 4/1/08 3:32 PM Page 36 Direct Substitution f ^ x h by simply substituting c for x and then evaluating Many times, you can find the limit lim x"c the resulting expression. In this section, you will see examples of how to determine the limits of polynomial, radical, and trigonometric functions, as well as how to determine the limit of a quotient of rational expressions. LIMIT OF A POLYNOMIAL FUNCTION Determine lim ^ x 2 + 3x h x"2 1 Begin with the original limit statement.
Lim x"2 lim x +3 + 5 x -2 $ x +3 - 5 x +3 + 5 =lim x"2 ^ x -2h` x +3 + 5j ^ x +3h -5 x"2 =lim x"2 3 Use direct substitution in the resulting expression. qxp 4/1/08 3:32 PM Page 48 Limits at Infinity: Horizontal Asymptotes This section discusses the behavior of the graph of a function as x approaches ± ϱ, in other words, limits at infinity. Definition of a Horizontal Asymptote The line y = L is a horizontal asymptote f(x) y = f(x) of the graph of y = f(x) if either lim f ^ x h =L or lim f ^ x h =L .