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Math II > Ch. 5 > Flashcards

Ch. 5 Flashcards

1
Q

discrete exponential growth

A

represented by points

Ex: most species only reproduce once a year

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2
Q

continuous exponential growth

A

represented by a line or curve etc.

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

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3
Q

exponential function

A

y = abx

y - amount after x period

a - intial value

b - growth factor

x - time period

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4
Q

how to write exponential functions and predict

A
  1. 9 mill ppl in 1980, increasing 1.7% each yr
    a. write an exponential function
    b. when will pop reach 4.5 mill?
    a.
  2. 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
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5
Q

exponential growth

A

y = abx

b > 1

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6
Q

exponential decay

A

y = abx

0 < b < 1

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7
Q

negative exponents

A

b-x = (1/b)x

Ex: y = 52(6/5)-x = 52(5/6)x

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8
Q

half-life questions

A

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

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9
Q

fractional exponent rules

A

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

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10
Q

solving exponent algebraic equations

A

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

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11
Q

e

A

the number e is an irrational # that = approx. 2.718281828 <- this is also the limit of the sequence: (1+1/1)1, (1+1/2)2, (1+1/3)3, …

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12
Q

logistic growth function

A

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

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13
Q

solving e problems

A

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

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14
Q

inverse functions

A

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)

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15
Q

graphing inverse functions

A

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)

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16
Q

deciding existence of inverse functions

A

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-

  1. Just like vertical line tests, u can use the horizontal line test to see whether a function has an inverse (original)
17
Q

writing inverse equations

A

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
Q

logarithmic function

A

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

x = ba → a = logbx

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
Q

logarithm

A

(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
Q

common logarithm

A

a log w/ base 10

common log of x written as log x

x = 10a → a = log x

21
Q

natural logarithm

A

a log w/ base e

usually written as ln x

x = ea → a = ln x

22
Q

evaluating logs

A

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
Q

solving logs

A

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
Q

properties of logs

A

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

25
Q

Product of Logarithms Property

A

logbMN = logbM + logbN

26
Q

Quotient of Logarithms Property

A

logbM/N = logbM - logbN

27
Q

Power of Logarithms Property

A

logbMk = k logbM

28
Q

writing in terms of logM and logN

A

a. logM2N3

logM2 + logN3 = 2logM + 3logN

b. log(3[M/N4)

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

29
Q

simplifying logs

A

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
Q

log and exp word problems

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

L1= 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)twith 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)t= 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

31
Q

Change of Base Property of Logs

A

For all pos. #’s with b and c not equaling 1,

logbM = (logcM)/(logcb)

32
Q

using change of base prop.

A

Evaluate log38to three dec. places

log38 = log8/log3 = abt 1.893

33
Q

graphing logs

A

Use a graphing calc to graph y = log6x

  1. Graph y = log6x or y = logx/log6
    1. You know that log66 = 1, so use TRACE to confirm by going to x=6

Math II (17 decks)

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