Exponential and logarithmic functions ppt

exponential function transformation ppt and exponential functions powerpoint algebra and introduction to exponential functions ppt
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Published Date:26-07-2017
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Exponential functions MATH 1510 Functions of the form Lili Shen x f(x) = 2 Exponential Functions are called exponential functions. When the variable is in the exponent, even a small change in the variable can cause a dramatic change in the value of the function: 3 f(3) = 2 = 8; 10 f(10) = 2 = 1024; 20 f(20) = 2 = 1; 048; 576: 2 As a comparison, for the function g(x) = x we have 2 2 2 g(3) = 3 = 9; g(10) = 10 = 100; g(20) = 20 = 400:Comparing exponential and power functions MATH 1510 Lili Shen 8 x f(x) = 2 Exponential Functions 2 g(x) = x 6 4 2 0 0 1 2 3Comparing exponential and power functions MATH 1510 Lili Shen 25 x f(x) = 2 Exponential Functions 2 g(x) = x 20 15 10 5 0 0 2 4 6Comparing exponential and power functions MATH 1510 Lili Shen 1;000 x f(x) = 2 Exponential Functions 2 g(x) = x 800 600 400 200 0 0 5 10 15 20Exponential functions MATH 1510 Lili Shen Using the language of calculus, this property of exponential Exponential Functions functions may be expressed as n x lim = 0 x x1 a for all a 1 and n2N, no matter how large n is or how small a is (as long as a 1). For example, 100 x lim = 0: x x1 1:01Irrational exponents MATH 1510 Lili Shen To study exponential functions, we must define what we x mean by the exponential expression a when x is any real Exponential Functions number. m Recall that for any a 0 and q = 2Q with n 0, n p m n m n a = a : x To define a when x is irrational, we approximate x by rational numbers; that is, we find an infinite sequence of rational numbersfqg with q q (n1), and define n n x q n a = lim a : n1Irrational exponents MATH 1510 Lili Shen For example, since Exponential Functions  = 3:1415926535:::  is an irrational number, we successively approximate a by the following rational powers: 3:1 3:14 3:141 3:1415 3:14159 a ; a ; a ; a ; a ;::: Intuitively, we can see that these rational powers of a are  getting closer and closer to a . It can be shown in calculus that there is exactly one number that these powers  approach, which is defined as the value of a .Laws of exponents MATH 1510 The following laws of exponents are still true when the Lili Shen exponents are real numbers: Exponential Functions Proposition x y x+y (1) a a = a . x a xy (2) = a . y a x y xy (3) (a ) = a . x x x (4) (ab) = a b .   x x a a (5) = . x b b   x x a b (6) = . x b a y x a b (7) = . x y b aExponential functions MATH 1510 Lili Shen Exponential Definition Functions The exponential function with base a is defined for all real numbers x by x f(x) = a ; where a 0 and a =6 1. x It is easy to see that the expression a makes sense for all x2R only when a 0, and we assume a =6 1 to eliminate the trivial case.Calculating exponentials MATH 1510 Lili Shen Exponential Exponentials are usually denoted by the symbol Functions (Shift+6 in the standard keyboard) in computers. For example, typing 3pi in Google and you will get  3  31:5442807002:Graphs of exponential functions MATH 1510 Lili Shen Exponential Functions Graphs of exponential functions have an easily recognizable shape. Example Draw the graph of each function. x (1) f(x) = 3 .   x 1 (2) g(x) = . 3Graphs of exponential functions MATH 1510 Solution. Lili Shen x y f(x) = 3 Exponential Functions 1 x 4 g(x) =( ) 3 3 2 1 x 2 1 1 2Graphs of exponential functions MATH 1510 Lili Shen Exponential Functions It can be inferred from the graphs that the graph of g can be obtained from the graph of f by reflecting in the y-axis. This is true since   x 1 1 x g(x) = = = 3 = f(x): x 3 3Graphs of exponential functions MATH 1510 More graphs of exponential functions are sketched below: Lili Shen Exponential y 1 x Functions y = ( ) 2 1 x y = ( ) 10 x 2 y = 10 x y = 2 1 x 2 1 1 2Graphs of exponential functions MATH 1510 Lili Shen The graph of every exponential function passes through the 0 point (0; 1) because a = 1 for any a =6 0. Exponential Functions x If 0 a 1, the function f(x) = a decreases rapidly. x If a 1, the function f(x) = a increases rapidly. y = 0 (the x-axis) is a horizontal asymptote for every x exponential function f(x) = a . This is because x if a 1, then a 0 as x1; x if 0 a 1, then a 0 as x1. x x Since a 0 for all x2R, the function f(x) = a has domainR and range (0;1).Graphs of exponential functions MATH 1510 Proposition Lili Shen For any a 0 with a =6 1, the exponential function Exponential Functions x f(x) = a has domainR and range (0;1). The line y = 0 (the x-axis) is a horizontal asymptote of f. The graph of f has one of the following shapes: y y 2 2 1 1 x x 2 1 1 2Identifying graphs of exponential functions MATH 1510 Example Lili Shen x Find the exponential function f(x) = a whose graph is Exponential Functions given. (1) y (2) y 8 30 (2; 25)  6 20 4 10 2 1 (3; ) 8 x x  2 2 4 2Identifying graphs of exponential functions MATH 1510 Lili Shen Exponential Functions Solution. 2 x (1) Sincef(2) = a = 25, it follows thata = 5. Sof(x) = 5 . 1 1 3 (2) Since f(3) = a = , it follows that a = . So 8 2   x 1 f(x) = . 2Compound interest MATH 1510 Lili Shen Exponential functions have important applications in Exponential calculating compound interests. If an amount of money P, Functions called the principal, is invested at an interest rate i per time period, then after one time period the interest is Pi, and the amount of money becomes A = P +Pi = P(1+i): 1 If the interest is reinvested, then the new principal is P(1+i), and the amount after another time period becomes 2 A = A (1+i) = P(1+i) : 2 1Compound interest MATH 1510 Lili Shen In general, after k periods the amount of money becomes Exponential Functions k A = P(1+i) : Note that this is an exponential function with base 1+i. If the annual interest rate is r and the interest is compounded n times per year, then in each time period the interest rate is r i = ; n and there are nt time periods in t years.

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