MATH 122 Fall 2015 Review Exam (section 1)

  1. The concentration of carbon dioxide, \( C=f(t) \), in the atmosphere, in parts per million (ppm), is a function of years, t, since 1960. Interpret \( f(40)=370 \).

  2. For the function \( f(x)=3x+3 \), find \( f(7) \).

  3. Annual sales of CDs have declined since 2000. Sales were 752.5 million in 2000 and 435.6 million in 2010.

    (a) Find a formula for annual sales, S, in millions of music CDs, as a linear function of the number of years, t, since 2000.

    (b) Use the formula to predict music CD sales in 2015 in million of CDs.

  4. Find the slope and y-intercept of the line of the equation \( 12x = 6y + 4 \).

  5. The table below shows attendance at the Atlanta Braves baseball games from 2001 to 2006.
    \begin{equation*} \begin{array}{|c||c|c|c|c|c|c|} \hline \text{Year} & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \text{Att.} & 20.72 & 21.15 & 22.53 & 22.31 & 22.42 & 22.43 \\ \hline \end{array} \end{equation*}

    Find the average rate of change in attendance from 2001 to 2006.

  6. Find the Relative Change of a population for the following changes:

    (A) From 500 to 2000

    (B) From 10000 to 5000

    (C) From 100,000 to 15,000

  7. Fixed Cost: 19,000 (dollars). Variable Cost: 14 (dollars/shirt). Revenue: 32 (dollars/shirt). Compute cost function, revenue function, profit function, graph and find break-even point.

  8. Find the relative rate of change in the price of a $44.99 pair of shorts if the sale price is $29.45.

  9. The population, P, in millions of Nicaragua was 6.3 million in 2006 and growing at an annual rate of 3.9% Let t be time in years since 2006.

    (a) express P as a function \( P=P_0 A^t \).

    (b) express P as an exponential function using base e.

    (c) compare the annual and continuous growth rates.

  10. The EPA invested a spill of radioactive iodine. The radiation level at the site was 1.6 milirems per hour. The level of radiation decays at a rate of \( k= -0.05 \).

    • the level of radiation 24 hours later?

    • number of hours until the level of radiation reached the max acceptable limit (0.6 milirems per hour).

  11. The quantity of the radioactive element mercury, m, is decaying exponentially at a continuous rate of 0.62% per year. What is the half life?

  12. You have your choice of receiving $1 million in four yearly installments of $250,000 each year starting now, or a lump-sum payment of $920,000 now. Assuming a 6% interest rate per year, compounded continuously, and ignoring taxes, which option would produce the higher value?

  13. Are the following functions power functions? If it is a power function, write it in the form \( y=kx^p \) and give the values of k and p.
    \begin{equation*} y=(4x^5)^2, \qquad y=\frac{9}{\sqrt[4]{x}}, \qquad y=3x^5+4 \end{equation*}
  14. The cost of producing q items is given by, \( C(q)= 0.04q^3+200q+750 \). Find \( C(40), MC(40) \), and explain what each one means.

  15. Compute the derivative of the following functions:

    (a) \( \displaystyle{f(x) = \big( 3x^{1/3} + \ln x \big)^{5/3} \big( e^{3x}-2x^5 \big)} \)

    (b) \( \displaystyle{f(x) = \sqrt{3x^2 + \ln x}} \)

    (c) \( \displaystyle{f(x) = \frac{4x^2 + \ln x}{2-4x^{4/3}+3^x}} \)

    (d) \( \displaystyle{f(x) = 150^{2x-1}} \)

    (e) \( \displaystyle{f(x) = \ln (2x^2+3x+3e^2 - 2)} \)

  16. Write an equation for a graph obtained by vertically stretching the graph of \( y=x^3 \) by a factor of 3 followed by a vertical upward shift of 2 units.

  17. In 2015, there were 543 million cans of soup being produced in the U.S. Also, in the same year, canned soup production is increasing by 36 million cans/year. If t is years since 2000, compute the relative rate of change in the year 2015 of canned soup production and interpret what it means.

  18. Estimate the Instantaneous Rate of Change for \( P= 150(1.4)^t \) at \( t=3 \).
The section corresponding to the third part of the test is very weak, and not representative of what will be in the final exam. There are so many different antiderivatives that you have not explored, and put too much emphasis on applications of optimization. Of all problems on optimization, I have only chosen 2 (the two most representative or original). The rest, will not receive any credit. There are also questions that belong in the fourth part of the course. I did not count those questions either, and will not receive any extra credit.
  1. Find the inflection point(s) of the function \( f(x)=24x^3+12x^2-11x \).

  2. Find the antiderivative of the following functions:

    (a) \( \displaystyle{\int \frac{3}{x} + e^{4x}\, dx} \)

    (b) \( \displaystyle{\int x \big( 8^{5x^2-57} \big)\, dx} \)

  3. Use the first derivative to identify each critical point as a local maximum, a local minimum or neither, for the function \( f(x)= x^4 - 2x^2 +7 \).

  4. The billion dollar shoe company All Star has been in business for decades. Numbers have changed drastically over the years and now accountants need to re-crunch numbers to determine how many pairs of shoes are needed to be produced to result in the maximum amount of profit for the company. Having said that, the company’s revenue and cost levels can be determined by: \( R(q)= 5q-.003q^2, C(q)= 300 + 1.1q \), where q represents units, and is equal to a positive value no greater than 1000 (\( 0 \leq q \leq 1000 \)). Given the previous information, determine how many pairs of shoes should be sold to allow the company to make the maximum amount of profit.

  5. Find the global maximum and global minimum of the function \( f(x) = x^3 - 6x^2 \) over the interval \( -2 \leq x \leq 10 \).

  6. At a price of $100 per concert ticket, a band attracts 250 buyers. Every $5 decrease in price attracts and additional 25 buyers.

    (a) Find the demand equation.

    (b) Express revenue as a function of price.

    (c) What price should the band charge per ticket to maximize revenue?

The problems from team #4 finally came! The choice of questions is also weak, with several people asking the same topic. I wish you'd had picked some Riemann sums, for example, or computation and applications of _total change_.
  1. Find the consumer surplus for the demand curve \( p = 100 - q^2/2 \) when \( q = 4 \).

  2. Find the area under the graph of \( f(x) = x^2+4 \) between \( x = 1 \) and \( x = 5 \).

  3. Find the area between the graphs of \( y = 7x-x^3 \) and \( y = \tfrac{1}{2}\sqrt{x} \) for \( x \geq 0 \).

  4. Compute the are of the region bounded by the graphs of \( x=1, x=5, y=x, y=4x-3 \).

  5. The population of Columbia, SC can be modeled by the function \( P = f(t) = 250 (1.025)^t \), where P is in thousands of people, and t is in years since 2000. Compute the average population of Columbia between the years 2003 and 2013.

  6. The value V of a lamp worth $140 in 1990 increases by 20% per year. Find the average value of the lamp between 2000 and 2015.

  7. After a foreign substance is introduced, the rate at which the white blood cells increase is found by \( r(t) = \dfrac{t}{t^2+1} \), with t being time in minutes. How many while blood cells will there be in 5 minutes?