Xa xa for any real number b b 7 0 lim b xa f x b c 6

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (x) =b c 6. lim f (x) d xSa A =b. 10. For any real number b, 0 6 b 6 1, or b 7 1, lim log b (f (x)) = log b A lim f (x) B = log b A; A 7 0. x:a x:a 7. 1 1 For any positive real number n, lim n = 0 and lim n = 0. x: q x x: -q x (If x is negative, x n does not exist for certain values of n, so the second limit is undefined.) Let p (x) and q(x) be polynomials, q(x) Z 0. To find log b x If x, a, and b are positive, a Z 1, b Z 1, then log a x = . log b a ln x In particular, log a x = . ln a LOGARITHMIC EQUATIONS For x 7 0, y 7 0, b 7 0, b Z 1, if x = y, then log b x = log b y. And if log b x = log b y, then x = y. lim x: q 1. 2. p(x) p(x) or lim : x : - q q(x) q(x) Divide the numerator and denominator by x raised to the highest power of x appearing in either polynomial. Then find the limit of the result from Step 1 by using the rules for limits, including the rules 1 1 lim n = 0 and lim n = 0. x: q x x: -q x more® FIRST DERIVATIVE TEST Let c be a critical number for a function. Suppose that f (x) is continuous on (a, b) and differentiable on (a, b) except possibly at c and that c is the only critical number on (a, b). 1. f (c) is a relative maximum if f ¿ changes from positive to negative at x = c. 2. f (c) is a relative minimum if f ¿ changes from negative to positive at x = c. Absolute Extrema of a Function f on an Interval [a, b] Chain Rule: If y = f (u) and u = g(x) so that y = f (g(x)), dy dy = dx du # d du or [ f (g(x))] = f ¿ (g(x)) # g ¿ (x) . dx dx Absolute extrema only occur at critical values of f or at the endpoints of the interval, a or b. dx [e ] = ex dx dx 9. [a ] = a x # ln a dx d g(x) 10. [a ] = (ln a)a g(x) # g ¿ (x) dx g ¿ (x) d 11. C ln ƒ g(x) ƒ D = dx g(x) 8. FINDING ABSOLUTE EXTREMA FOR f ON [a, b ] 1. 2. 3. 4. USES OF THE FIRST DERIVATIVE • • • • • 2 f (c) is a relative maximum on (a, b) if f (x) … f (c) for all x in (a, b). 2. f (c) is a relative minimum on (a, b) if f (x) Ú f (c) for all x in (a, b). The term relative extremum refers to either a m...
View Full Document

This document was uploaded on 03/08/2014 for the course QSCI 291 at University of Washington.

Ask a homework question - tutors are online