Logarithm and Exponential Derivatives and Integrals

Transcription

1 Logarithm and Exponential Derivatives and Integrals James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 3, 2013

2 Outline 1 Exponential Functions: Integrals and Derivatives Derivative s Integral s 2 Logarithm Functions: Integrals and Derivatives Derivative s Integral s

3 Abstract This lecture is going to talk the derivatives and integrals of natural logarithm function ln(x) and the exponential function exp(x) = e x.

4 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),

5 Exponential Functions: Integrals and Derivatives If x and y are positive numbers then exp(x + y) = exp(x) exp(y), If x and y are positive numbers then exp(x y) = exp(x) exp( y) = exp(x) exp(y), If x and y are positive numbers then. (exp(x)) y = exp(xy),

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

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

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

9 Exponential Functions: Integrals and Derivatives Derivative s Find the derivative of exp( 3x). d (exp( 3x)) = exp( 3x) ( 3) dx Find the derivative of exp(5x). d (exp(5x)) = exp(5x) 5 dx

10 Exponential Functions: Integrals and Derivatives Derivative s 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). 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

11 Exponential Functions: Integrals and Derivatives Derivative s 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.

12 Exponential Functions: Integrals and Derivatives Integral s Find the integral of exp(x 2 + x + 1) (2x + 1). exp(x 2 + x + 1) (2x + 1) = exp(u) du, use substitution u = x 2 + x + 1 = exp(u) + C = exp(x 2 + x + 1) + C

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

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

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

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

17 Exponential Functions: Integrals and Derivatives Integral s 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

18 Exponential Functions: Integrals and Derivatives Integral s 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.

19 Logarithm Functions: Integrals and Derivatives We know ln is a continuous function of x for positive x, lim x ln(x) =, lim x 0 ln(x) =, ln(1) = 0, ln(e) = 1, ( ln(x) ) = 1 x,

20 Logarithm Functions: Integrals and Derivatives and we know If x and y are positive numbers then ln(xy) = ln(x) + ln(y), If x and y are positive numbers then ln( x ) = ln(x) ln(y), y If x and y are positive numbers then ln(x y ) = y ln(x).

21 Logarithm Functions: Integrals and Derivatives We can say more about the derivative of the logarithm function. Then ln( x ) is nicely defined at all x not zero. ln( x ) = { ln(x) if x > 0 ln( x) if x < 0 If x is negative, using the chain rule we have d dx (ln( x)) = 1 x d dx ( x) = 1 x ( 1) = 1 x

22 Logarithm Functions: Integrals and Derivatives Thus, d (ln x )) = dx = 1 x We conclude that { d dx { (ln(x)) if x > 0 1 d dx (ln( x)) if x < 0 = x if x > 0 1 x if x < 0 if x is not 0 d dx (ln( x )) = 1 x, if x 0 So the antiderivative of 1/x is ln( x ) + C.

23 Logarithm Functions: Integrals and Derivatives We have two new rules: Let s do some examples. d dx (ln( u(x) ) = 1 u(x) u (x) 1 du = ln( u ) + C u

24 Logarithm Functions: Integrals and Derivatives Derivative s Find the derivative of ln(x 2 + x + 1). d ( ln(x 2 + x + 1) ) = dx 1 x 2 (2x + 1) + x + 1

25 Logarithm Functions: Integrals and Derivatives Derivative s Find the derivative of ln(tan(x)). d dx (ln(tan(x))) = 1 tan(x) sec2 (x) Now the natural logarithm here is only defined when tan(x) is positive. We usually just assume that we are only interested in evaluating the expression ln(tan(x) for such x; i.e. by writing ln(tan(x), we are tacitly assuming that tan(x) is positive. Another way of looking at this, is that the domain of the function ln(tan(x) is the set of x where tan(x) is positive. However, it is best to train ourselves to think about the restrictions on the domain without being so explicit!

26 Logarithm Functions: Integrals and Derivatives Derivative s Find the derivative of ln(5x). d dx (ln(5x)) = 1 5x 5 = 1 x Here the implied domain of ln(5x) is all positive x. We can again use the properties of ln to get this result another way. d dx (ln(5x)) = d dx (ln(x)) + d dx (ln(5)) = 1 x

27 Logarithm Functions: Integrals and Derivatives Derivative s Find the derivative of ln(5x 2 + 6x + 9). d ( ln(5x 2 + 6x + 9) ) = dx 1 5x 2 (10x + 6). + 6x + 9 Find the derivative of ln(5x 3 + 6x 2 + 9x 25). d ( ln(5x 3 + 6x 2 + 9x 25) ) = dx 15x x + 9 5x 3 + 6x 2 + 9x 25.

28 Logarithm Functions: Integrals and Derivatives Derivative s Homework Find the derivative of ln(t 2 + 1) Find the derivative of ln(t 3 + t 2 + 5) Find the derivative of ln(y 2 + 3y 12) Find the derivative of ln(x x 3 23x + 100) Find the derivative of t ln(t) t.

29 Logarithm Functions: Integrals and Derivatives Integral s Find the integral 1 (2x + 1). x 2 +x+1 1 x 2 (2x + 1) = + x u du, use substitution u = x 2 + x + 1 = ln( u ) + C = ln( x 2 + x + 1 ) + C

30 Logarithm Functions: Integrals and Derivatives Integral s Find the integral 1 tan(x) sec2 (x) dx. 1 tan(x) sec2 (x) dx = 1 u du, use substitution u = tan(x); = ln( u ) + C = ln( tan(x) ) + C

31 Logarithm Functions: Integrals and Derivatives Integral s Find the integral e t 1 + e t dt. e t 1 + e t dt = 1 u du, use substitution u = 1 + e t ; du = e t dt = ln( u ) + C = ln( 1 + e t ) + C, but 1 + e t is always positive = ln(1 + e t ) + C

32 Logarithm Functions: Integrals and Derivatives Integral s Find the integral 5x 4 + x 2 dx. 5x 4 + x 2 dx = u du, use substitution u = 4 + x 2 ; = 5 2 ln( u ) + C = 5 2 ln( 4 + x 2 ) + C, but 4 + x 2 is always positive = 5 2 ln(4 + x 2 ) + C

33 Logarithm Functions: Integrals and Derivatives Integral s Find the integral tan(w)dw. tan(w)dw = = sin(w) cos(w) dw 1 u ( du), use substitution u = cos(w); du = sin(w)dw = ln( u ) + C = ln( cos(w) ) + C

34 Logarithm Functions: Integrals and Derivatives Integral s Homework Find the integral ln(t) t dt ( let u = ln(t)) Find the integral 2t/(t 2 + 4) dt Find the integral 2s 2 /(s 3 + 9) ds Find the integral 8z z 3 + 9z + 18 dz Find the integral 40z z z 8z 5 + 9z z dz Find the integral 2 0 t 4t dt.