Practice Final

Practice Final

Key

Problem 1

Let

f(x)=(x-2)/(x+3), g(x) = 1/(x+4)

A. Find the domain of f o g(x).

B. Find f o g(x).

Problem 2

Let

f(x) = 2x/(3x-1)

A. Prove that f is a 1-1 function.

B. Find the inverse of f.

Problem 3

Sketch the graph of

Problem 4

Put into exponential form to find the exact value of

log_5 (1/root(125))

Problem 5

Write the expression as a sum and/or difference of logarithms. Express powers as factors and simplify.

ln[ root(x-1) / (x^2 e^x) ]

Problem 6

Solve the following equations.

B. (e^x - e^-x)/2 = 5

Problem 7

Brad, who is 21 years old, wants to invest $30,000 in a mutual fund that historically has grown 8% per year compounded continuously. Brad figures he needs $1,000,000 to retire, at what age can Brad retire assuming that the mutual fund grows as it has historically throughout the investment period?

Problem 8

The expected logistics growth model for the quagga muscle if it finds its way into Lake Tahoe is

P(t) = 30000000/(1+10000000e^(-1.2t))

A. Determine the carrying capacity for Lake Tahoe.

B. What is the growth rate of the muscle?

C. Determine the initial population size.

D. Find the population 5 years after the muscle arrives.

E. When will the population be 20,000,000?

Problem 9

Find the vertex, focus, and directrix of the parabola. Then sketch the graph.

Problem 10

The aphelion of Jupiter is 507 million miles and the distance from the center of its elliptical orbit to the Sun is 23.2 million miles. Find the perihelion and the mean distance. Then write an equation for the orbit of Jupiter around the Sun.

Problem 11

Find an equation for the hyperbola with center at (2,-1), focus at (2,4) and vertex at (2,2). Then graph the equation.

Problem 12

Solve the system of equations using an augmented matrix. If the system has no solution say that it is inconsistent.

3x-y = 10, x+2y = 1

Problem 13

Solve for x.

Det Matrix: [ 2 1 x, 3 x 1, 2 1 0 ] = 4x

Problem 14

Let

A = Matrix[2 7, 1 3], B = Matrix[0 2 5, -1 3 -4]

Find the each of the following that is defined. If it is not defined, explain why.

AB, BA, A^-1, B^-1, A + 2B, 2A + A^-1

Problem 15

Graph the system of inequalities.

y >= x^2 + 4

Problem 16

A casino needs to determine how many slot machines and how many table games to put in its 6000 square foot room so that it staff of 64 employees can manage the games. Every ten machines takes one employee and 120 square feet of floor space, while each table requires two employees and takes us 100 square feet of floor space. Each group of 10 slot machines brings in a profit of $4,000 and each table brings in a profit of $6,000. How many slot machines and how many tables should the casino host in order to maximize profit?

Problem 17

Write down the n^th term of the sequence suggested by the pattern

1/4, -8/9, 27/16, -64/25, 125/36,...

Problem 18

Given that

sum(1) = n, sum(k) = n(n+1)/2, sum(k^2) = n(n+1)(2n+1)/6

Find

sum(3-2k+k^2)

Problem 19

The 12^th term of an arithmetic sequence is 7 and the 18^th term is 31. Find the 5th term.

Problem 20

The new city theater will have 18 seats in the first row and each successive row will contain two additional seats. If there are 35 rows in the theater, how many seats will there be total?

Problem 21

Suppose that the second term of a geometric sequence is 2/27 and the fifth term is 6. Find the sum of the first ten terms.

Problem 22

The government is giving a $1,000 stimulus check to each citizen. They figure that everyone will spend half of their money and 12% will come back to the government as taxes. The vendors that collect the spent money will in turn spend half of their sales and 12 percent of the amount spent will go to taxes. Then of this spent money, half will be spent again and 12% will come back as taxes. If this continues forever, how much tax revenue will the government receive in total?

Problem 23

Use Mathematical Induction to prove that the statement is true for all natural numbers:

1+5+5^2+...+5^(n-1) = 1/4 (5^n - 1)

Problem 24

Prove that if a sequence is defined recursively by

Then the closed form of the sequence is

Problem 25

Expand using the Binomial Theorem

Problem 26

Find the coefficient of x^3/2 in the expansion of

( 3root(x) + 1/root(x) )^7

Problem 27

How many different 12-letter words (real or imaginary) can be formed from letters in the word "TAHOECOLLEGE"?

Problem 28

A poker hand consists of five cards from a 52 card deck. The same five cards dealt in different orders count as the same poker hand. How many different poker hands are there?