If you get 2c on the first day of school, 4c on the second day and 8c on the third day, what should you get for attending on the 30th day?
y = 2x
y = 230
Estimate 240 in terms of a power of 10
210 ≃ 103
(210)4 ≃ (103)4
240 ≃ 1012
Estimate the power to which 10 must be raised to yield 2
103 ≃ 210
(103)1/10 ≃ (210)1/10
103/10 ≃ 2
If bx = by and b ≠ 1 then?
x = y
Solve (3√2)2
(32)√2
9√2
Solve 4x = 8
(22)x = 23
22x = 23
2x = 3
x = 3/2
Let b denote an arbitrary positive constant other than 1. The exponential function with base b is defined by the equation:
y = bx
In many applications, functions of the form y = abkx, where a, b, and k are constants, with b ≠ 1, are also called _______ ______.
exponential functions
State 3 key features of the graph y = 2x
The domain is the set of all real numbers and the range is a set of all positive real numbers
The y-intercept of the graph is 1. The graph has no x-intercept
For x > 0, the function increases or grows very rapidly. For x < 0, the graph rapidly approaches the x-axis is a horizontal asymptote for the graph
Solve y = (1/2)x
1x/2x = 1/2x = 2-x
What do you do when asked to graph 2-x - 2 = 0, what do you do?
figure out the x intercept by substituting x for 0
figure out the y intercept by substituting y for 0
plot
Let a be any real number. Then the instantaneous rate of change of the function f(x) = ex at x = a is ___. In other words, at each point on the graph of f(x) = ex, the instantaneous rate of change is just the _______ of the point.
exy-coordinate
At x = 2 the instantaneous rate of change of f(x) = ex is ____
e2
T or F: f(x) = bx is one to one (by the _______ test) and therefore possesses an inverse
True
horizontal line test
Define logbx and state an example
The exponent that a number/base "b" must be raised to, in order to yield x
Ex] log28 = 3
y = logbx is the ________ form and is equivalent to x = by which is the ______ form
logarithmic form
exponential form
The function y = logbx grows or increases very ______.
Slowly
10 = log21024 means we must go out beyond _____ on the _axis before the curve y = log2x reaches a height of 10 units
1000
x-axis
Evaluate log432 (just write the next step that will lead to the answer)
4y = 32
(22)y = (25)
2y = 5
y = 5/2
logex =
ln x
State the inverse of y = ln x
y = ex
Find the domain of f(x) = log2(12-4x)
12-4x > 0
x < 3
(-∞,3)
Rewrite e2t-5 = 5000 in logarithmic form
ln5000 = 2t - 5
State the equation that defines the Richter magnitude M of an earthquake
M = log10 (A/A0)
ln(0)
log(0)
undefined
undefined
The natural log or the log of any negative number is ______
*P and Q are assumed to be positive in properties 2-6
Change of base formula
logax = logbx/logba
How would you find out if x = 1 or x = 2 is a root of the equation (ln x)2 = 2lnx
Plug each in as 1 and then 2, if the answers are equivalent on both sides of the equation it is a root
ln (x)2 = 2 ln x holds for every value of x in the domain of the function _____. This example serves to remind us of the difference between a ______ ______ and an _____
y = ln x
conditional equation
identity
_______ is true for all values of the variable in its domain. For example, the equation ln(x2) = 2 ln x is an _______; it is true for every positive real number x. In contrast, a ______ ________ is true only for some (maybe even none) of the values of the variable.
An identity
identity
conditional equation
Using the properties of logarithms to solve logarithmic equations may introduce extraneous solutions that do not check in the original equation. Explain
The logarithmic function requires positive inputs, but in solving an equation, we may not know ahead of time the sign of an input involving a variable so we must ALWAYS check any candidates for solutions obtained in this manner
1a)If p < q then bp < bq
1b)Conversely, if bp < bq then p < q
Assuming b is a positive constant greater than 1, how does this work with logs
2a) If p < q then logbp < logbq
2b) If logbp < logbq, then p < q
Whats the first thing to do when asked "solve the following inequality a) ln(2 - 3x) ≤ 1"
We need to determine the domain of the function y = ln(2 - 3x). Since the inputs for all natural logarithm and logarithms must be positive, we require 2 - 3x > 0 and therefore x < 2/3
To increase a given quantity by 15%, we multiply the quantity by _____. Similarly, to increase a quantity by 30%, we would multiply by _____ and so on.
1.15
1.30
Suppose that you place $1000 in a savings account at 10% interest compounded annually (explain). Interest compounded in this manner is called _____ _____. The original deposit of $1000 is called the _______ (__). The interest rate, expressed as a decimal is denoted by r which is ____ in this example and A is the _____ in the account at any given time
This means that at the end of each year, the bank contributes to your account 10% of the amount that is in the account at that time
compound interest
Principal (P)
.10
amount
"t" not only represents time, it also represents ____________
number of compoundings
Compound Interest Formula (interest compounded annually)
A = P(1 + r)t
*If a principal of P dollars is invested at an annual rate r that is compounded annually. The formula denotes amount (A) after t years
When asked how much time it would take for a bank's annual compound interest to double the principal amount of money in an account, and you calculate 12.7 years, what is the correct answer?
After 13 years the initial amount will be more than doubled
General formula for periodic interest rate
r/n, where r is the annual interest rate and n is the number of times per year that the interest is compounded
Compound Interest Formula (interest compounded n times per year)
A = P(1 + r/n)nt
*If a principal of P dollars is invested at an annual rate r that is compounded n times per year. The formula denotes amount (A) after t years
In the compound interest formula (interest compounded n times per year) explain r/n and the exponent nt
r/n is the periodic interest rate
nt represents the total number of compoundings
If the interest for the year under quarterly compounding was $1103.81 - $1000 = $103.81, then $103.81 is ______% of $1000. We can then say in this case, the effective rate or _____ _____ of interest is _______. The given rate of 10% per annum (ex3 in 5.6) is called the _____ ______ or _____ ____
10.381%
annual yield
10.381%
nominal rate or annual rate
Compound Interest Formula (interest compounded continuously)
A = Pert
*If a principal of P dollars is invested at an annual rate r that is compounded continuously. The formula denotes amount (A) after t years
Doubling time
the amount of time required for a given principal to double *also applies to exponential population growth
Explain why doubling time does not depend on Principal (P)
A = Pert
*A = 2P
2P = Pert
2 = ert
ln 2 = rt
t = ln2/r
What is the formula for Doubling time (T2) under continuous compounding
Doubling time = T2 = ln2/r
Exponential growth refers specifically (in sciences) to which type of growth? State an example
Growth governed by functions of the form y = aebx, where a and b are positive constants. For example A = Pert
Exponential decay refers specifically (in sciences) to? State one example
Decreases or decay governed by functions of the form y = aebx, where a is positive and b is negative
Ex: population growth, global warming and radio activity
Function for population model of bacteria in ideal condition ie unlimited food and space
Growth law: N(t) = N0ekt
N(t) is the population at time t
k is the growth constant, a positive constant related to (but not equal to) the growth rate of the population
N0 is also a constant; it represents the size of the population at time t = 0
Formula for percentage increase in a population over a given time interval
percentage increase = (change in population/initial population) * 100
What is the average relative growth rate (aka relative growth rate) of a population
percent increase per hour
Two characteristics of exponential growth
relative growth rates are all equal (for the same function)
relative growth rate is approximately equal to the rate constant "k" (provided k is close to 0)
The half-life of a radioactive substance
The time required for half of a given sample to disintegrate.
The half-life is an intrinsic property of the substance (meaning)
