By Dale W. Johnson
Calculus can try out the bounds of even the main complex math scholars. This visible, easy-to-follow e-book deconstructs advanced mathematical strategies in a manner that’s infinitely more uncomplicated to know. 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 recommendations capability you don’t need to kind via long educational textual content, and you may consult with the Appendix for a glance at universal differentiation ideas, integration formulation, and trigonometric identities.
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Extra info for Teach Yourself VISUALLY: Calculus
X " 0- x " 0+ 1 From the graph at right, you can see that lim =-1 x"0 and lim f ^ x h =0. qxp 4/1/08 3:22 PM Page 19 1 chapter An Introduction to Limits 2 Notice that since the left-sided and right-sided limits are different, lim f ^ x h does not exist. x"0 ONE-SIDED LIMITS FOR A POLYNOMIAL FUNCTION 3 2 For the function f ^ x h = x - x - 3x, find lim f ^ x h and lim f ^ x h . 4 6 x " -2 x " -2 - + 1 From the graph at the right, you can f(x) 3 23 see that lim f ^ x h =3 2 and 3 x " -2 2 lim f ^ x h =3 3 .
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]. Case I: f has both a minimum and a maximum value on (a,b). Case II: f has an extreme value at a and another extreme value in (a,b). f(x) f(x) max max min x a b x a b min Case III: f has an extreme value at b and another extreme value in (a,b). Case IV: f has the same maximum and minimum value.
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. 2 Substitute 2 for x, then simplify.