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Properties of Common Functions Properties of ln x 1. The domain is the set of all positive real numbers x > 0. 2. The range is the set of all real numbers °1 < y < 1. 3. Algebraic properties: If a and b are any positive real numbers, and r is any real number, then (a) ln 1 = 0 (b) ln ab = ln a + ln b (Product rule) (c) ln ab = ln a ° ln b (Quotient rule) (d) ln ar = r ln a (Power rule) (e) ln a1 = ° ln a 4. DiÆerentiation and Integration: Z d 1 1 ln x = , dx = ln |x| + C, dx x x and Z ln x dx = x ln x ° x + C Properties of ex 1. The domain of the exponential function is the set of all real numbers, °1 < x < 1. 2. The range of the exponential function is the set of all positive real numbers y > 0. 3. The exponential function is the inverse of the natural logarithm function. This means eln x = x for all x > 0, and ln ex = x for all x 2 R. 4. Algebraic Properties: (a) e0 = 1 (b) ex+y = ex ey (c) ex°y = ex /ey (d) e°x = 1/ex 5. DiÆerentiation and Integration: d x e = ex dx and 1 Z ex dx = ex + C. Properties of Common Functions Trigonometric Functions 1. Identities (a) Pythagorean: sin2 µ + cos2 µ = 1, tan2 µ + 1 = sec2 µ (b) Parity: sin(°µ) = ° sin µ, cos(°µ) = cos µ (c) Addition Formulas: i. sin(µ + ¡) = sin µ cos ¡ + cos µ sin ¡ ii. cos(µ + ¡) = cos µ cos ¡ ° sin µ sin ¡ (d) Product Formulas: i. sin µ sin ¡ = 12 (cos(µ ° ¡) ° cos(µ + ¡)) ii. cos µ cos ¡ = 12 (cos(µ ° ¡) + cos(µ + ¡)) iii. sin µ cos ¡ = 12 (sin(µ + ¡) + sin(µ ° ¡)) (e) Amplitude-Phase Shift Formulas: p i. A cos µ + B sin µ = C cos(µ ° ¡), where C = A2 + B 2 and tan ¡ = B/A p ii. A cos µ + B sin µ = C sin(µ + ¡), where C = A2 + B 2 and tan ¡ = B/A 2. DiÆerentiation and Integration R d sin x = cos x dx R d cos x = ° sin x dx d tan x = sec2 x dx sin x = ° cos x + C cos x = sin x + C d sec x = sec x tan x dx 2