155b help sheet

155b help sheet - Second Semester Calculus Math 155B Inverse functions A function f is one-to-one(1-1 on an interval if f x1 = f x2 x1 = x2 for all

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Second Semester Calculus, Math 155B Inverse functions : A function f is one-to-one (1-1) on an interval if 2 1 2 1 ) ( ) ( x x x f x f = = for all values 1 x and 2 x in the interval. The graph of a 1-1 function must pass the Horizontal Line Test : No horizontal line can intersect the graph in more than one point. If a function f is 1-1, it has an inverse function defined by y x f x y f = = - ) ( if only and if ) ( 1 . This means that if f takes x to y , then 1 - f takes y to x . The domain of f is the range of 1 - f , and the range of f is the domain of 1 - f . A function and its inverse have the following cancellation relations: x x f f = - )) ( ( 1 and y y f f = - )) ( ( 1 Exponential functions : If n is a positive integer and a is a positive real number, then we know that times ... n n a a a a = , n n a a 1 = - , and 1 0 = a . For a rational number q p r = with 0 q , q p q p r a a a = = . If x is a real number, define r x r x a a = lim for rational numbers r . Properties of exponential functions : The exponential function x a x f = ) ( is a continuous function with domain ) , ( -∞ and range ) , 0 ( . If x and y are real numbers, y x y x a a a + = y x y x a a a - = xy y x a a = ) ( x x x b a ab = ) ( The number e : The number e is defined to be that unique positive number with the property that the slope of the tangent line to the graph of x e y = is 1 when x = 0. The value of e is approximately 2.718. The function x e x f = ) ( is called the natural exponential function . This function is differentiable with x x e e dx d = ) ( and (if u is a differentiable function of x) dx du e e dx d u u = ) ( The associated indefinite integral is C e dx e x x + = .
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Logarithm functions : The general exponential function x a x f = ) ( is increasing if 1 a and decreasing if 1 0 < < a . This means the function is 1-1 and has an inverse. The inverse of the exponential function x a x f = ) ( is called the logarithm function to the base a , which is defined by y a a x x y = = if only and if log . Since the range of the exponential function is ) , 0 ( , the domain of the logarithm function is ) , 0 ( , which means that the logarithm function is defined only for positive values of x . Cancellation properties : y a y a = log and x a x a = log Graphs of logarithm functions : If 1 a , the logarithm function x y a log = is a continuous, increasing function with domain ) , 0 ( and range ) , ( -∞ . Natural logarithm : The logarithm function to the base e is called the natural logarithm function : x x e log ln = . The natural logarithm function is related to the general logarithm function by the formula a x x a ln ln log = . If x and y are positive numbers and r is any real number, then y x xy ln ln ) ln( + = y x y x ln ln ln - = ( 29 x r x r ln ln = The natural logarithm function and the natural exponential functions are inverses of each other, so x e x = ) ln( for all x and x e x = ln for
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This note was uploaded on 02/27/2009 for the course MATH 155b taught by Professor Staff during the Fall '08 term at Vanderbilt.

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155b help sheet - Second Semester Calculus Math 155B Inverse functions A function f is one-to-one(1-1 on an interval if f x1 = f x2 x1 = x2 for all

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