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The Exponential Function Integer Powers Fractional Powers and Irrational Powers Properties of Powers The Mathematical Constant e The Exponential Function

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General Powers? We all know that 2 2 = 2 2 = 4. Starting Question What means 2 ? Functions/Elementary Functions/The Exponential Function by M. Seppälä

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General Power Functions For a number a, one defines the positive integer powers of a recursively by setting 1.a 1 = a, 2.a 2 = a a, 3.a n+1 = a a n for n > 1. Negative integer powers are defined by setting for n = 1, 2, …. Assuming that a 0, one defines a 0 by setting a 0 = 1. Warning 0 0 is undefined. One cannot assign a value to 0 0. This is an example of indeterminates. In computer mathematics systems such expressions are called NaNs, short for Not a Number. Functions/Elementary Functions/The Exponential Function by M. Seppälä

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Roots Let n be a positive integer, and a > 0. Let b be a positive number such that b n = a. Requiring b be positive makes it uniquely defined. Definition The number b is the positive nth root of the positive number a. Warning Even roots of negative numbers are not real numbers, i.e., there are no real numbers whose even power would be negative. Notation If n is a positive even number and a positive, the equation b n = a has always two solutions b and -b. If n is odd, the solution is unique and positive. Functions/Elementary Functions/The Exponential Function by M. Seppälä

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Fractional and Irrational Powers Let p be an integer, and q a positive integer. For a positive number a define the fractional power a p/q by setting Definition The precise definition of a ρ for irrational numbers ρ can be given by approximating an irrational number ρ by rational numbers p/q, and then approximating the irrational power a ρ by the rational powers a p/q. In this way a x can be defined for all real numbers x assuming that a > 0. We will not go into more details at this point. These considerations can be made rigorous. The general power can also be defined using integration. We will later use this method to give a precise definition of general powers. Functions/Elementary Functions/The Exponential Function by M. Seppälä

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Properties of Powers 1 1 2 2 3 3 4 4 If a > 1, x > y a x > a y. 5 5 If 0 y a x < a y. 6 6 The following properties of powers follow from the definition. Here we assume that a is positive. Functions/Elementary Functions/The Exponential Function by M. Seppälä

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Exponential Functions General exponential functions are functions of the form f(x) = a x for some positive number a. The figure on the right shows the graphs of the functions 1.y = (1/2) x, the red curve, 2.y = 1 x, the black line, 3.y = (3/2) x, the blue curve, and 4.y = (5/2) x, the green curve. Observe that, if a > 1, the function y = a x is increasing. The greater the number a is, the faster the values of the function y = a x increase. If 0 < a < 1, the function y = a x is decreasing. If a 1 and a > 0, the function y = a x is monotonic bijection between real numbers and positive real numbers, y: +, y = a x. Functions/Elementary Functions/The Exponential Function by M. Seppälä

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a=1/2 a=1 a=3/2 a=5/2 The Number e Definition e 2.718281828 The slope of a tangent line is the tangent of the angle at which the tangent line intersects the x-axis. The mathematical constant e is defined as the unique number for which the slope of the tangent line of the graph of the function e x at x = 0 is 1. From the picture it is obvious that, as the parameter a grows, the slope of the line, tangent to the graph of the function a x at the point (0,1), grows. Functions/Elementary Functions/The Exponential Function by M. Seppälä

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The Exponential Function Definition The function y = e x is the exponential function. Since the mathematical constant e is greater than 1, the exponential function is an increasing bijection between the set of real numbers and the set of positive real numbers. The exponential function is an important basic function in calculus. Here we have defined the basic mathematical constant e using a rather heuristic geometric condition. The constant e can be defined more rigorously by consider the expression for large values of the integer n. One can show that, as n grows, the numbers (1 + 1/n) n approach the mathematical constant e. We will make these remarks precise later. Functions/Elementary Functions/The Exponential Function by M. Seppälä

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