Equations
Calculate the total amount due after 22 months compounded quarterly on a loan of $1200 with a simple interest charge of 5.5%
$1320.38
What are the units for the average rate of change?
output units / input unit
What is the definition of an inflection point?
The point where the concavity switches.
In order for a limit to exist, what must be true?
The right hand limit must equal the left hand limit.
Find the time it takes for the investment to double. You invest $500 compounded continuously at a rate of 4.5%. Write your answer in years and months.
15 years and 5 months
The percent of Bebo users in the world between x ages 17 and 33 can be modeled as: u(x)=18-6.74ln x percent.
Calculate the average rate of change and the percent change for users between the ages of 20 and 30.
Average rate of change = -0.273 % / age
Percent change - 124.717%
What are the three directions for a function on a graph?
1. increasing
2. decreasing
3. constant
What are the conditions for continuity?
1. f(c) exists
2. lim f(x) exists
as x -> c
3. lim f(x) = f(x)
x -> c
According to the social security advisory board, the number of workers per beneficiary of the social security program was 4.7 in 1996 and is projected to decline by 2.38% each year through 2030. Write a model for this situation (all four components).
f(x) = 4.7(0.9762)x workers per beneficiary is the projected decline of workers x years after 1996 where 0 < x < 34.
The annual number of passengers going through Hartsfield-Jackson Atlanta International Airport between 2000 and 2008 can be modeled as p(t)=-0.102t3+ 1.39t2 – 3.29t + 79.25 where output is measured in million passengers and t is the number of years since 2000.
a. Find the percentage rate of change at t = 6.
2.713%
Tell which types of functions have no concavity, one concavity, and two concavities.
None - linear
One - exponential, logarithmic, and quadratic
Two - logistic and cubic
Graph on document...
What is the limit at x = 1? Is the function continuous?
Yes the function is continuous
The limit is 6.
The percentage of vacant rental housing units in the US during the 4th quarter for the selected years between 2000 and 2004 is given in the table.
Year: 2000 2001 2002 2004
% Rental Vacancy: 7.8 8.8 9.3 10.0
a. Align the data since 1999. Write a completely defined logarithmic model.
b. When you are talking about 2003, are you using interpolation or extrapolation?
a. f(x) = 7.818 + 1.361lnx
b. interpolation
Describe the trends in tangent lines in relation to how it touches or crosses the graph.
Concave up - tangent line is below the graph
Concave down - tangent line is above the graph
Inflection point - tangent line cuts through the graph at the point
Tell which types of functions have asymptotes. Include how many and which type.
Exponential - 1 horizontal asymptote
Logarithmic - 1 vertical asymptote
Logistic - 2 horizontal asymptotes
(draw graph on board) Is this graph continuous? Does the limit exist?
Function is not continuous.
The limit does not exist.
The average number of viewers for a movie on a Saturday night t hours since 4 PM is V(t) viewers. The average amount of tips generated by v viewers is P(v) dollars.
a. Write how these functions can be composed. What are the new input and output units?
b. Write a statement for the new function complete with function notation and input and output units and descriptions.
a. P(v(t)) // input unit - hours since 4 pm // output unit - dollars
b. P(v(t)) dollars gives the average amount of tips generated by viewers of a movie t hours since 4 pm on a Saturday night.
According to the US Census Bureau, S(x)=-0.461x2+8.656x+202.357 billion dollars gives the annual profit of US pharmacies and drug stores, where x is the number of years since 2007, 0<x<5.
a. Write a sentence of interpretation for S’(3)=5.89.
In 2010, the annual profit of US pharmacies and drug stores was increasing by 5.89 billion dollars / year.
Use numerical estimation to find the limits of the function as x approaches ∞ and -∞.
F(x) = (500) / (1+2e0.5x)
Limit as approaches ∞ = 0
Limit as approaches -∞ = 500
Describe the end behavior of the following graph of f.
(graph on document)
lim f(x) = 3
x --> ∞
lim f(x) = 0
x --> -∞