Defining Exponential Functions and Exponential Derivatives and Integrals

Transcription

1 Defining Exponential Functions and Exponential Derivatives and Integrals James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 19, 2014

2 Outline 1 The Exponential Function 2 Exponential Functions: Integrals and Derivatives Derivative Examples Integral Examples

3 Abstract This lecture develops the exponential function and discusses integrals and derivatives for it.

4 The Exponential Function Let s backup and talk about the idea of an inverse function. Say we have a function y = f (x) like y = x 3. Take the cube root of each side to get x = y 1/3. Just for fun, let g(x) = x 1/3 ; i.e., we switched the role of x and y in the equation x = y 1/3.

5 The Exponential Function Let s backup and talk about the idea of an inverse function. Say we have a function y = f (x) like y = x 3. Take the cube root of each side to get x = y 1/3. Just for fun, let g(x) = x 1/3 ; i.e., we switched the role of x and y in the equation x = y 1/3. Now note some interesting things: f (g(x)) = f (x 1/3 ) = g(f (x)) = g(x 3 ) = ( x 1/3 ) 3 = x ( x 3 ) 1/3 = x.

6 The Exponential Function Now the function I (x) = x is called the identity because it takes an x as an input and does nothing to it. The output value is still x. So we have f (g) = I and g(f ) = I. When this happens, the function g is called the inverse of f and is denoted by the special symbol f 1.

7 The Exponential Function Now the function I (x) = x is called the identity because it takes an x as an input and does nothing to it. The output value is still x. So we have f (g) = I and g(f ) = I. When this happens, the function g is called the inverse of f and is denoted by the special symbol f 1. Of course, the same is true going the other way: f is the inverse of g and could be denoted by g 1. Another more abstract way of saying this is f 1 (x) = y f (y) = x.

8 The Exponential Function Now look at next Figure. We draw the function x 3 and its inverse x 1/3 in the unit square. We also draw the identity there which is just the graph of y = x; i.e. a line with slope 1.

9 The Exponential Function Now look at next Figure. We draw the function x 3 and its inverse x 1/3 in the unit square. We also draw the identity there which is just the graph of y = x; i.e. a line with slope 1. If you take the point (1/2.1/8) on the graph of x 3 and draw a line from it to the inverse point (1/8, 1/2) you ll note that this line is perpendicular to the line of the identity. This will always be true with a graph of a function and its inverse.

10 The Exponential Function Now look at next Figure. We draw the function x 3 and its inverse x 1/3 in the unit square. We also draw the identity there which is just the graph of y = x; i.e. a line with slope 1. If you take the point (1/2.1/8) on the graph of x 3 and draw a line from it to the inverse point (1/8, 1/2) you ll note that this line is perpendicular to the line of the identity. This will always be true with a graph of a function and its inverse. Now also note that x 3 has a positive derivative always and so is always increasing. It seems reasonable that if we had a function whose derivative was positive all the time, we could do this same thing. We could take a point on that function s graph, say (c, d), reverse the coordinates to (d, c) and the line connecting those two pairs would be perpendicular to the identity line just like in our figure. So we have a geometric procedure to define the inverse of any function that is always increasing.

11 The Exponential Function

12 The Exponential Function What about ln(x)? It has derivative 1/x for all positive x, so it must be always increasing. So it has an inverse which we can call ln 1 (x) which is called the exponential function which is denoted by exp(x).

13 The Exponential Function What about ln(x)? It has derivative 1/x for all positive x, so it must be always increasing. So it has an inverse which we can call ln 1 (x) which is called the exponential function which is denoted by exp(x). The inverse is defined by the if and only relationship (ln) 1 (x) = y ln(y) = x

14 The Exponential Function What about ln(x)? It has derivative 1/x for all positive x, so it must be always increasing. So it has an inverse which we can call ln 1 (x) which is called the exponential function which is denoted by exp(x). The inverse is defined by the if and only relationship (ln) 1 (x) = y ln(y) = x or, using the exp notation exp(x) = y ln(y) = x.

15 The Exponential Function What about ln(x)? It has derivative 1/x for all positive x, so it must be always increasing. So it has an inverse which we can call ln 1 (x) which is called the exponential function which is denoted by exp(x). The inverse is defined by the if and only relationship (ln) 1 (x) = y ln(y) = x or, using the exp notation exp(x) = y ln(y) = x. A little thought tells us the range of ln(x) is all real numbers as for x > 1, ln(x) gets as large as we want and for 0 < x < 1, as x gets closer to zero, the negative area 1 1/tdt approaches. x

16 The Exponential Function What about ln(x)? It has derivative 1/x for all positive x, so it must be always increasing. So it has an inverse which we can call ln 1 (x) which is called the exponential function which is denoted by exp(x). The inverse is defined by the if and only relationship (ln) 1 (x) = y ln(y) = x or, using the exp notation exp(x) = y ln(y) = x. A little thought tells us the range of ln(x) is all real numbers as for x > 1, ln(x) gets as large as we want and for 0 < x < 1, as x gets closer to zero, the negative area 1 1/tdt approaches. x By definition then ln( exp(x) ) = x for < x < ; ie for all x. exp( ln(x) ) = x for all x > 0.

17 The Exponential Function We know ln( exp(x)) = x. Take the derivative of both sides: ( ) ( ln( exp(x) = x) = 1 Using the chain rule, for any function u(x), ( ) 1 ln( u(x) = u(x) u (x). So ( ln( exp(x) ) = ( 1 exp(x)). exp(x) ( 1 Using this, we see exp(x) exp(x)) = 1 and so ( exp(x)) = exp(x). This is the only function whose derivative is itself!

18 Exponential Functions: Integrals and Derivatives We know exp is a continuous function of x for all x, lim x exp(x) =, lim x exp(x) = 0, exp(0) = 1, ( exp(x) ) = exp(x),

19 Exponential Functions: Integrals and Derivatives We have two new rules: d dx ( exp( u(x) ) ) = exp( u(x) ) u (x) exp(u) du = exp(u) + C Let s do some examples.

20 Exponential Functions: Integrals and Derivatives Derivative Examples Example Find the derivative of exp(x 2 + x + 1). Solution Example d ( exp(x 2 + x + 1) ) = exp(x 2 + x + 1) (2x + 1) dx Find the derivative of exp(tan(x)). Solution d dx (exp(tan(x))) = exp(tan(x)) sec2 (x)

21 Exponential Functions: Integrals and Derivatives Derivative Examples Example Find the derivative of exp(exp(x 2 )). Solution Example d ( exp(exp(x 2 )) ) = exp(exp(x 2 )) exp(x 2 ) 2x dx Find the derivative of exp( 5x 3 + 2x 8). Solution d ( exp( 5x 3 + 2x 8) ) = exp( 5x 3 + 2x 8) ( 15x 2 + 2) dx

22 Exponential Functions: Integrals and Derivatives Derivative Examples Example Find the derivative of exp( 3x). Solution Example d (exp( 3x)) = exp( 3x) ( 3) dx Find the derivative of exp(5x). Solution d (exp(5x)) = exp(5x) 5 dx

23 Exponential Functions: Integrals and Derivatives Derivative Examples Before we go any further, you need to know that historically, many people got tired of writing exp all the time. They shortened the notation as follows: exp(u(x)) = e u(x). We will explain why this is ok to do in a bit. Sometimes it is easier to read a complicated exponential expression using the e u(x) notation and sometimes not. Judge for yourselves. 1 d ( ) e x2 +x+1 = e x2 +x+1 (2x + 1) dx 2 d ( e 3t ) = e 3t ( 3) dt 3 d ( e 5t ) = e 5t 5 dt

24 Exponential Functions: Integrals and Derivatives Derivative Examples Homework Find the derivative of exp( 3t) Find the derivative of exp(t 2 + 2t 4) Find the derivative of e 1/t Find the derivative of e t /(e t + e t ) Find the derivative of (t 2 + 2t 8) e 5t Find the derivative of e 4t Find the derivative of e 3t Find the derivative of e 18t.

25 Exponential Functions: Integrals and Derivatives Integral Examples Example Find the integral of exp(x 2 + x + 1) (2x + 1)dx. Solution exp(x 2 + x + 1) (2x + 1)dx = exp(u) du, use u = x 2 + x + 1 = exp(u) + C = exp(x 2 + x + 1) + C

26 Exponential Functions: Integrals and Derivatives Integral Examples Example Find the integral of exp(tan(x)) sec 2 (x) dx. Solution exp(tan(x)) sec 2 (x) dx = exp(u) du, use u = tan(x); = exp(u) + C = exp(tan(x)) + C

27 Exponential Functions: Integrals and Derivatives Integral Examples Example Find the integral of exp(t 2 ) 3t dt. Solution exp(t 2 ) 3t dt = exp(u)3(du/2), use u = t 2 ; = (3/2) exp(u) du = (3/2) exp(u) + C = (3/2) exp(t 2 ) + C

28 Exponential Functions: Integrals and Derivatives Integral Examples Example Find the integral of exp( 3t) dt. Solution exp( 3t) dt = exp(u) du/( 3), (u = 3t) use u = 3t; = 1 exp(u) du 3 = 1 3 exp(u) + C = 1 3 exp( 3t) + C

29 Exponential Functions: Integrals and Derivatives Integral Examples Example Find the integral of exp(5t) dt. Solution exp(5t) dt = = 1 5 exp(u) du/(5), (u = 5t) exp(u) du = 1 5 exp(u) + C = 1 5 exp(5t) + C

30 Exponential Functions: Integrals and Derivatives Integral Examples You should see the problems above using the other notion also. Here they are: 1 e t2 3t dt = e u 3(du/2), (u = t 2 ) = (3/2) e u du = (3/2) e u + C = (3/2) e t2 + C 2 e 3t dt = e u du/( 3), (u = 3t) = 1 e u du 3 = 1 3 eu + C = 1 3 e 3t + C

31 Exponential Functions: Integrals and Derivatives Integral Examples Homework Find the integral exp( 3t) dt Find the integral exp(t 2 + 2t 4) (2t + 2)dt Find the integral e t3 t 2 dt Find the integral t e 5t2 dt Find the integral e 14t dt Find the integral 5 1 e 33t dt Find the integral 2 0 e4t dt.