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1

discrete exponential growth

represented by points

Ex: most species only reproduce once a year

2

continuous exponential growth

represented by a line or curve etc.

Ex: over-lapping generations. some species bred thruout the year

3

exponential function

y = abx

y - amount after x period

a - intial value

b - growth factor

x - time period

4

how to write exponential functions and predict

2.9 mill ppl in 1980, increasing 1.7% each yr

a. write an exponential function

b. when will pop reach 4.5 mill?

a. 

  1. find growth factor b

pop @ end of yr = 100% of pop @ start + 1.7% of pop @ start

pop @ end of yr = 101.7% pop @ start of year

b = 1.017

  1. y = abx

y = (2.9)(1.017)x

b. 

  1. Use a graph
    1. Graph the equation y = (2.9)(1.017)x
    2. x-value of 0 = 1980, x-value of 26 = 2006
  2. Use recursion
    1. Enter initial amount 2.9 & repeatedly multiply by growth factor (1.017)
    2. Count # times pressed Enter (260
    3. 1980 + 26 = 2006

5

exponential growth

y = abx

b > 1

 

6

exponential decay

y = abx

0 < b < 1

7

negative exponents

b-x = (1/b)x

Ex: y = 52(6/5)-x = 52(5/6)x

8

half-life questions

1200 ppl; every 1/2 hour, 1/2 ppl leave

1 half-life → (1200)(1/2)

2nd half-life → (1200)(1/2)(1/2)

f(x) = 1200(1/2)x

how many ppl after 3 hrs?

about 18

9

fractional exponent rules

a1/n = n[a      Ex: 41/3 = 3[4

am/n = (n[a)m or n[am      Ex: 53/4 = (4[5)3 or 4[53

IF N IS EVEN, MUST USE ABSOLUTE VALUES

10

solving exponent algebraic equations

Find the value of b when f(x) = 4bx and f(3/4) = 32

4b3/4 = 32

b3/4 = 8

b3/4(4/3) = 84/3
b = 84/3 = (3[9)4 = 24 = 16

11

e

the number e is an irrational # that = approx. 2.718281828   1, (1+1/2)2, (1+1/3)3, ...

 

12

logistic growth function

a function in which the rate of growth of a quantity slows down after initially increasing/decreasing exponentially

 

13

solving e problems

spread of flue in 1,000 ppl modeled by y = 1000/(1 + 990e-0.7x), where y = the # of ppl after x days

a. how many ppl after 9 days?

Graph the equation

Read y when x = 9 -> abt 355ppl

b. horizontal asymptotes?

Trace along same graph 

min. y-value gets close to but never reaches 0

max. y-value gets close to but never reaches 1000

y = 0, y = 1000

c. Estimate max # of ppl

max. of y = 1000 ppl

14

inverse functions

two functions f & g r inverse functions if g(b) = a whenever f(a) = b

graph = reflection of the graph of the original function over the line y = x

inverse of f(x) can be written f-1(x) or f-1 (although the exp. -1 usually means reciprocal, f-1(x) is NOT 1/f(x)

 

15

graphing inverse functions

a. y = 2x b. y = x2

1. Plot pts to graph each function. Then interchange the coordinates of the original points and plot these points.      

a. (1,2) → (2,1)

(-1, 1/2) → (1/2, -1)

b. (2,4) → (4,2)

(-2,4) → (4,-2)

 

16

deciding existence of inverse functions

an inverse is a function. if a reflection of a function over the line y = x isn't a function, then the inverse doesn't exist

a. y = 2x b. y = x2

1. Use the vertical line test to decide whether a reflection is the graph of a function (inverse)

-or-

2. Just like vertical line tests, u can use the horizontal line test to see whether a function has an inverse (original)

17

writing inverse equations 

Ex: Write an equation for the inverse function of f(x) = 2/3x - 4

  1. f(x) = 2/3x - 4
  2. y = 2/3x - 4
  3. x = 2/3y - 4
  4. x + 4 = 2/3y
  5. (3/2)x + (3/2)4 = (3/2)(2/3)y
  6. 3/2x + 6 = y
  7. f-1x = 3/2x + 6

so just interchange x & y, and solve for y

18

logarithmic function

the inverse of the exponential function f(x) = bx and written as f-1(x) = logbx

x = ba   →   a = logb

x on outside, b in middle, a in center

base of a log can be any pos. # except 1

log4(-2) is undefined! can't have neg log

19

logarithm

(of any pos. real # x) the exponent a when u write x as a power of a base b

x = ba  iff  a = logbx

base of a log can be any pos. # except 1

20

common logarithm

a log w/ base 10

common log of x written as log x

x = 10a   →   a = log x 

21

natural logarithm

a log w/ base e

usually written as ln x

x = ea   →   a = ln x

22

evaluating logs

Evaluate each. (Find a). Round decimal answers to the nearest hundredth

a. log464

log464 = log443 → a = 3; has to be 43 cuz base = 4

log464 → 4a = 64 → a = 3

b. log(1/10,000)

log(1/10000) = log10-4 → a = -4

c. ln5.3

use calc; = abt 1.67

d. log145

use calc; = abt 2.16

23

solving logs

when solving these, remember that x isn't necessarily the x in the formula. it can be a, b, or x

solve logx81 = 2

logb81 = 2

b2 = 81

b = 9

Solve. round decimal answers to nearest hundredth

a. 10x = 15

10a = 15

a = log 15 = abt 1.18

b. ea = 29

ea = 29

a = ln 29 = abt 3.37

c. log2x + log2(x-2) = 3

log2x + log2(x-2) = 3

log2x(x-2) = 3

x(x-2) = 23

x2 - 2x = 8

x2 - 2x - 8 = 0

(x+2)(x-4) = 0

x + 2 = 0   or   x - 4 = 0

x = -2 x = 4

check for extraneous solutions! Sub possible solutions into the original equation to be sure they're not extraneous

x = -2 is undefined, cuz no neg logs log2(-2)

 

24

properties of logs

M, N, and P are pos. #'s w/ b not equaling 1, and k is any real #

25

Product of Logarithms Property

logbMN = logbM + logbN

26

Quotient of Logarithms Property

logbM/N = logbM - logbN

27

Power of Logarithms Property

logbMk = k logM

28

writing in terms of logM and logN

a. logM2N3

logM2 + logN3 = 2logM + 3logN

b. log(3[M/N4)

log3[M - logN4 = logM1/3 - logN4 = 1/3logM - 4logN

 

29

simplifying logs

ln18 - 2ln3 + ln4

ln18 - ln32 + ln4  → power prop.

ln18 - ln9 + ln4

ln18/9 + ln4  → quotient prop.

ln2 + ln4

ln(2•4)  → product prop.

ln8

30

log and exp word problems

  1. L = 10log(I/I0) What is the change in loudness(L) when I is doubled?

L= original loudness   L2 = loudness after intensity is doubled

increase in loudness = L2 - L1

L2 - L1 = 10log(2I/I0) - 10log(I/I0)

10log(2•I/I0) - 10log(I/I0)

10(log2 + logI/I0) - 10log(I/I0)

10log2 + 10log(I/I0) - 10log(I/I0)

10log2 = abt 3

  1. A(t) = A0(0.883)with A(t) - amount present t thousand yrs after death, A0 - amount @ time of death, and t - amount time since death (in 1000's of yrs)

        found in 1968, died 8000 yrs ago, 37% present now; estimate age of the bone

A0(0.883)t = A(t)

A0(0.883)= 0.37A0

0.883t = 0.37

log(0.883)t = log0.37

t log 0.883 = log0.37

t = log0.37/log0.883

t = 8