The term relative extremum refers to either a maximum

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Unformatted text preview: aximum or a minimum. If f has a relative extremum at x = c, then c is a critical number of f (x). 1. dy d , [ f (x)], Dx [ f (x)] dx dx Constant Rule: If f (x) = k, then f ¿ (x) = 0. Power Rule: If f (x) = x n, then f ¿ (x) = nx n-1. d Constant Times a Function: [k f (x)] = k # f ¿ (x) dx Sum or Difference: If f (x) = u(x) ; v(x), then f ¿ (x) = u ¿ (x) ; v ¿ (x). Product Rule: If f (x) = u(x) # v(x), then f ¿ (x) = u(x)v ¿ (x) + v(x)u ¿ (x). u(x) Quotient Rule: If f (x) = and v(x) Z 0, then v(x) v(x)u ¿ (x) - u(x)v ¿ (x) . f ¿ (x) = [v(x)]2 then LIMITS AT INFINITY CHANGE OF BASE FOR LOGS RELATIVE EXTREMA Notation: f ¿ (x), lim 3f (x) ; g (x)4 = lim f (x) ; lim g (x) = A ; B x:a A critical number is any number c for which f ¿ (c) = 0 or f ¿ (c) is undefined. TECHNIQUES FOR FINDING DERIVATIVES x:a lim k # f (x) = k lim f (x) = kA 6. 5. ,!7IA3C1-dheeaa!:t;K;k;K;k more® x:a 3. PROPERTIES OF LOGARITHMIC FUNCTIONS log a (xy) = log a x + log a y x log a a b = log a x - log a y y log a x r = r log a x log a a = 1 x:a If k is a constant, then lim k = k. CRITICAL NUMBERS A function which describes the instantaneous rate of change (or slope of tangent line) of f (x) at any point x is called the derivative. f (x + h) - f (x) Derivative: f ¿ (x) = lim h:0 h Let a, A, and B be real numbers and let f and g be continuous functions such that lim f (x) = A and lim g (x) = B. For a 7 0, a Z 1, and x 7 0, y = log a x means a y = x. The logarithm, y, is the exponent to which you must raise a to produce x. 1. h:0 x:a 1. 2. f (a + h) - f (a) is h the instantaneous rate of change of f (x) at x = a. Instantaneous rate of change: lim lim f (x) only exists if lim- f (x) = lim+ f (x). RULES FOR LIMITS Suppose a function has a derivative at each point in the open interval. 1. If f ¿ (x) 7 0 for each x in the interval, then f (x) is increasing on the interval. 2. If f ¿ (x) 6 0 for each x in the interval, then f (x) is decreasing on the interval. 3. If f ¿ (x) = 0 for each x in the interv...
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