Exponential Functions

Transcription

1 10.1 Unlimited growth and doubling Bunnies A pair of bunnies makes 4 babies in 1 generation. Suppose a bunny weighs 1 kg, and the earth weighs kg. When will the bunnies weigh as much as the earth, if a generation is 3 months?

2 10.1 Unlimited growth and doubling Exponential Functions Definition An exponential function is a function of the form f (x) = a x, where a is some constant. Contrast: a power function is a function of the form f (x) = x a, where a is some constant. exponential: f (x) = 2 x f (x) = 5 x f (x) = 1 2 x = 0.5 x not exponential: f (x) = x 2 f (x) = x 5 f (x) = x = x 0.5 Exponential functions and power functions look similar, but they behave in fundamentally different ways.

3 10.1 Unlimited growth and doubling Graphing Exponential Functions 50 y y = 3 x y = 2 x y = ( ) 1 x 2 5 x

4 10.1 Unlimited growth and doubling Graphing Exponential Functions y = 2 x y = 3 x y = ( ) 1 x

5 10.1 Unlimited growth and doubling y y = b x y = c x y = a x x Order a, b, c, and 1 from smallest to largest. Order 1 a, b, c, and 1 from smallest to largest.

6 10.2 Derivatives of exponential functions and the function e x Derivatives of Exponential Functions y f (x) = a x 1 x small large Consider d dx {ax }. f (x) is always increasing, so f (x) is always positive. f (x) might look similar to f (x).

7 10.2 Derivatives of exponential functions and the function e x Exponential Functions d dx {ax } = lim h 0 a x+h a x h = lim h 0 a x a h a x h = lim h 0 a x (a h 1) h = a x lim h 0 (a h 1) h = a x (times a constant) Given what you know about d a h 1 dx {ax }, is it possible that lim =0? h 0 h A. Sure, there s no reason we ve seen that would make it impossible. B. No, it couldn t be 0, that wouldn t make sense. How could we find out what this limit is?

8 10.2 Derivatives of exponential functions and the function e x Exponential Functions In general, d dx {ax } = a x lim h 0 a h 1 h Euler s Number for any positive number a. e h 1 We define e to be the unique number satisfying lim = 1 h 0 h e (Wikipedia) Derivatives of Exponential Functions Using the definition of e, d dx {ex } = e x e h 1 lim = e x h 0 h In general, lim ah 1 = ln(a), so d dx {ax } = a x ln(a) [proof]

9 10.2 Derivatives of exponential functions and the function e x Quick Practice Things to Have Memorized When a is any constant, d dx {ex } = e x d dx {ax } = a x log e (a) Let f (x) = ex. When is the tangent line to f (x) horizontal? 3x 5 Horizontal tangent line slope of tangent line is zero

10 10.2 Derivatives of exponential functions and the function e x Evaluate d { dx e 3x }.

11 10.2 Derivatives of exponential functions and the function e x Example 1: According to the collision theory of bimolecular gas reactions, a reaction between two molecules occurs when the molecules collide with energy greater than some activation energy, E a, referred to as the Arrhenius activation energy. E a > 0 is constant for the given substance. The fraction of bimolecular reactions in which this collision energy is achieved is F = e Ea RT, where T is temperature (in degrees Kelvin) and R > 0 is the gas constant. Suppose that the temperature increases at some constant rate, C, per unit time. Determine the rate of change of the fraction F of collisions that result in a successful reaction.

12 10.3 Inverse functions and logarithms Invertibility: Grading Code Key message Good Job! Nice Idea! Algebra Mistake Creative Strategy! Write your Name Use a Logarithm Please don t submit papers with coffee stains code GJ NI AM CS WYN LOG CS The longer code is not uniquely translatable; viewed as a function, it is not invertible.

13 10.3 Inverse functions and logarithms Functions are Maps domain range f (x) f 1 (x)

14 10.3 Inverse functions and logarithms y x A. invertible B. not invertible

15 10.3 Inverse functions and logarithms Relationship between f (x) and f 1 (x) Let f be an invertible function. What is f 1 (f (x))? A. x B. 1 C. 0 D. not sure domain range f (x) 5 25 f 1 (x)

16 10.3 Inverse functions and logarithms Invertibility In order for a function to be invertible, different x values cannot map to the same y value. We call such a function one-to-one, or injective. Example 2: Suppose f (x) = x 3. What is f 1 (3)? (simplify your answer) What is f 1 (10)? (do not simplify) What is f 1 (x)?

17 10.3 Inverse functions and logarithms Example 3: Let f (x) = x 2 x. 1. Sketch a graph of f (x), and choose a domain over which it is invertible. 2. For the domain you chose, evaluate f 1 (20). 3. For the domain you chose, evaluate f 1 (x). 4. What are the domain and range of f 1 (x)? What are the (restricted) domain and range of f (x)?

18 10.3 Inverse functions and logarithms Domain and Range f (x) = x 2 x, domain: [ 1 2, ) f 1 (x) = x 2 domain of f (x) f (x) range of f (x) [ 1 2, ) [ 1 4, ) f 1 (x) range of f 1 (x) domain of f 1 (x)

19 A.13 Logarithms Exponents and Logarithms f (x) = e x f 1 (x) = ln(x) = log(x) So, ln(e x ) = x and e ln x = x. x e x ln fact e fact 0 1 ln(1) = 0 e 0 = 1 1 e ln(e) = 1 e 1 = e 1 1 e ln( 1 e ) = 1 e 1 = 1 e n e n ln(e n ) = n e n = e n ln(1) = ln(e) = ln( 1 e ) = ln(e n ) =

20 A.13 Logarithms y y = e x y = x (1, e) ( 1, 1/e) (0, 1) (1, 0) (1/e, 1) (e, 1) y = ln(x) x

21 A.13 Logarithms Logs of Other Bases log = A. 0 B. 8 C. 10 D. other log 2 16 = A. 1 B. 2 C. 3 D. other

22 A.13 Logarithms Logarithm Rules Let A and B be positive, and let n be any real number. ln(a B) = ln(a) + ln(b) Proof: ln(a B) = ln(e ln A e ln B ) = ln(e ln A+ln B ) = ln(a) + ln(b) ln(a/b) = ln(a) ln(b) Proof: ln(a/b) = ln ( e ln A e ln B ) = ln(e ln A ln B ) = ln A ln B ln(a n ) = n ln(a) Proof: ln(a n ) = ln (( e ln A) n) = ln ( e n ln A ) = n ln A Simplify into ( a) single logarithm: 10 f (x) = ln x ln x + ln(10 + x)

23 A.13 Logarithms Base Change In general, for positive a, b, and c: b log b (a) = a ln(b log b (a) ) = ln(a) log b (a) ln(b) = ln(a) log b (a) = ln(a) ln(b) log b (a) = log c(a) log c (b)

24 A.13 Logarithms Base Change In general, for positive a, b, and c: log b (a) = log c(a) log c (b) Suppose your calculator can only compute logarithms base 10. What would you enter to calculate ln(17)? Suppose your calculator can only compute natural logarithms. What would you enter to calculate log 2 (57)? Suppose your calculator can only compute logarithms base 2. What would you enter to calculate ln(2)?

25 Chapter 2: Differentiation 2.10 The Natural Logarithm Differentiating the Natural Logarithm Calculate d dx {ln x}. One Weird Trick: x = e ln x d dx {x} = d dx 1 = e ln x ln {e x} 1 x = d {ln x} dx d dx {ln x} = x d {ln x} dx Derivative of Natural Logarithm d dx {ln x} = 1 x d dx {ln x } = 1 x (x > 0) (x 0)

26 Chapter 2: Differentiation 2.10 The Natural Logarithm Derivative of Natural Logarithm d dx {ln x } = 1 x (x 0) Differentiate: f (x) = ln x 2 + 1

27 Chapter 2: Differentiation 10.4 Applications of the logarithm Manipulating Exponential Equations and Logarithms Example 4: (a) Write these expressions as exponential functions with base e: 2 x, 1 5 x. (b) Find the derivative of f (x) = a x, where a is a positive constant, using the fact that d dx [ex ] = e x. (c) (d) Find the zero(es) of the function e ax e bx2, where a and b are constants. Suppose the quantity (in µg) of a radioactive isotope at time t (measured in years) is given by Q(t) = 50e t 700. When is Q(t) = 40 µg? How long does it take for half the substance to decay?

28 Chapter 2: Differentiation 10.4 Applications of the logarithm Log scale in action:

29 Chapter 2: Differentiation 10.4 Applications of the logarithm World Populations (wikipedia) 80 million GER TUR 60 million 40 million 20 million CAN AUS 2 million 95 thousand GAM SEY

30 Chapter 2: Differentiation 10.4 Applications of the logarithm World Populations log scale CHI IND USA GER TUR CAN AUS log 10 ( ) 9.1 log 10 ( ) 9.1 log 10 ( ) 8.5 log 10 ( ) 7.9 log 10 ( ) 7.9 log 10 ( ) 7.6 log 10 ( ) 7.4 GAM log 10 ( ) 6.3 SEY log 10 ( ) 5.0

31 Chapter 2: Differentiation 10.4 Applications of the logarithm Decibels: For a particular measure of the power P of a sound wave, the decibels of that sound is: 10 log 10 (P) So, every ten decibels corresponds to a sound being ten times louder. A lawnmower emits a 100dB sound. How much sound will two lawnmowers make? A. 100 db B. 110 db C. 200 db D. other

32 Chapter 2: Differentiation 10.4 Applications of the logarithm Logarithmic Differentiation - A Fancy Trick Differentiate: f (x) = ( (x 15 9x 2 ) 10 (x + x 2 + 1) ) 5 (x 7 + 7)(x + 1)(x + 2)(x + 3)