Math 1A Fall06 - Vertical Line Test (Function Test) - A...

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Vertical Line Test (Function Test) - A curve in the xy plane is the graph of a function of x if and only if no vertical line intersects the curve more than once Manipulating a Function y = f(x) + c, shift c units upward y = f(x) - c, shift c units downward y = f(x - c), shift c units to right y = f(x + c), shift c units to left y = cf(x), stretch vertically by factor c y = (1/c)f(x), compress vertically by factor c y = f(cx), compress horizontally by factor c y = f(x/c), stretch horizontally by factor c y = -f(x), reflect about the x-axis y = f(-x), reflect about the y-axis Inverse Functions - Must pass horizontal line test (one-to-one) - Reflection off the line y = x - Domain of f -1 = range of f - Range of f -1 = domain of f Log Properties log a x = y a y = x log a (a x ) = x a logax = x log(1) = 0 log(10) = 1 ln(1) = 0 ln(e) = 1 a x = e xlna log a x = ln(x) / ln(a) Trig Graphs Trig Values Limits lim f(x) = L if and only if lim f(x) = L = lim f(x) x→a- x→a+ lim sin(θ) = 1 lim cos(θ) - 1 = 0 x→0 θ x→0 θ Definition of Derivation f′(x) = lim f(x + h) - f(x) h→0 h f′(x) = lim f(u) - f(x) u→x u - x Derivatives d / dx f(x)g(x) = f(x)g′(x) + g(x)f′(x) d / dx f(x) = g(x)f′(x) – f(x)g′(x) g(x) g(x) 2 d / dx a (x) = a (x) ln(a) d / dx e (x) = e (x)
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