2. Find the equation (in the form ) of a line through points (1, 7) and (-3,2)

3. The relationship between air temperature T (in ?F) and altitude h (in feet above seal level) is approximately linear up to an altitude of 20,000 feet. If the temperature at seal level is 60?, an increase of 5,000 feet in altitude lowers the air temperature to 18?.
a. Express T in terms of h, and sketch the graph on an hT- coordinate system.
b. Using the equation determine the altitude at which the temperature is 0 ?F?

4. A certain country taxes the first $20,000 of an individual's income at the rate of 15%, and all income over $20,000 at 20%. Find a piecewise-defined function T that specifies the total tax on an income of x dollars. ( A graph of the function is not necessary)

5. Graph the function . Identify any asymptotic behavior.

1. The exponential function for the amount of money accumulated in a mutual fund is , where P is the initial principal invested, r is the annual dividend rate and t is time in years. Suppose you intend to invest a $10,000 inheritance in a fund that has a historical interest rate of 9% (r = .09). Show the graph of the value of your investment over 15 years. Compute the value of the investment after 10 years using the function above. But with the Economy in the tank, the 9% may be grossly optimistic. Re-compute and graph the value using an interest rate of 4%. Find the difference in your earnings.

2. The Reliability function, , indicates the probability of survival over time and applies to many electronic products. You will note that this is an exponential function; t is time (in hours) and m is a parameter called the MTTF or Mean Tine To Failure. Suppose for your 4G Ipod the reliability function; m is 5,000 hours. Draw a graph of this curve ( 0 ? t ? 10,000), and describe it's features (i.e. increasing or decreasing, y-intercept, asymptotes). Compute is the probability of survival at the 5,000 hour point using the Reliability function above.

3. Evaluate the following using your Scientific Calculator (round your answer to 4 decimal places)

a) b) c) d) e)

4. Solve the following:
a) Write as a single logarithm

1. Solve the recurrence exactly and prove your solution is correct by induction.
T(1) = 1 , T(n) = 2T(n-1)+2n-1
T(1) = 1, T(n) = T(n-1)+3n-3
2. Give asymptotic bound for the following:
T(n) = 9T(n/3)+n2
T(n) = T(Sqrt(n))+1.

Define the logarithmic integral li(x) as the integral of the function 1/(log t) from t = 2 to t = x, where x > 2 and "log" denotes the natural logarithm.
(a) Determine constants A and B such that li(x) can be expressed in the following two forms:
(i) li(x) = x/(log x) + A + g(x), where g(x) is the integral of thefunction

1) Determine the order of the following expressions as ??0:
?(?(1-?) ), (4?)^2 ?, 1000?^(1?2), ln?(1+?), (1-cos??)/(1+cos?? ), ?^(3?2)/(1+sin?? ), ?^(3?2)/(1-cos?? ),
sech ^(-1) ?, exp?(tan?? ), ln[1+(ln?(1+2?))/(1-2?)], ln[1+(1+2?)/(?(1-2?))], ?_0^? exp?(-s^2)ds.
2) Arrange the following in descending order for ??0.

Evaluate the exponential equation for three positive values of x, three negative values of x, and at x=0. Show your work. Use the resulting ordered pairs to plot thegraph. Show your wok and state if the equation of the line is asymptotic to thegraph (if any).
y = 3-x
Evaluate the exponential equation for three positive v

Pick a rational function.
Here are some examples you can use:
y = (x+1)/(x-2), y = 3x/(x^2-1),
y = (2x-1)/4x, y = (x+3)/(x^2-1),
y = (x^2+1)/(x^2 -3),
y = (6x+1)/(x^2),
y = x^2/(x-3),
y = (3x-5)/(4x+7),
y = (x^2)/(x^3 - 1)
Find:
a) Vertical Asymptote (if any)
b) Horizontal Asymptote (if any)
c) Slant Asympt

For a "notch" filter with transfer function P(s) = [s^2 + 0.1s + 1]/[s^2 + 10.1s +1]
i) Sketch theasymptotic Bode magnitude plot on a piece of logarithmic graph paper
ii) Correct your asymptotic sketch at 0.5, 1 and 2 times the value of each corner frequency, and
iii) Make a reasonable sketch of the Bode phase plot on a sepa

1. Evaluate the exponential equation for three positive values of x, three negative values of x, and at x=0. Show your work. Use the resulting ordered pairs to plot thegraph. State the equation of the line asymptotic to thegraph (if any).
y = 3x - 4
2. Evaluate the logarithmic equation for three values of x that

Determine theasymptotic stability of the system x' = Ax where
A is 3 x 3 matrix
A = -1 1 1
0 0 1
0 0 -2
( first row is -1 1 1 second is 0 0 1 and third is 0 0 -2)
More specifically, what stability conclusion(s) can be drawn? ( Justify your answer)