Algebra Review and Initial Applications. Section 003

This is a good first review, although I am missing a few questions on solving exponential/logarithmic equations, or dealing with exponential functions in general. I would like to have seen also some questions exploring the relation between the constants k, r and a. Otherwise, this is a fine selection. 4/5

Algebra Review and Initial Applications. Section 019

Much better selection of problems, and all topics and skills have been properly addressed. I prefer the applications of the other section, but yours are not bad. 5/5

1. Compute the slope of the line that goes through the points \( (2,6) \) and \( (4,12). \)

2. The value of a washing machine, V (in dollars) is a function of the age of the washing machine, a in years. We have \( V=700-75a. \) Give the units and meaning of the slope.

3. Write a function obtained from the function \( f(x)=x^2 \) by shifting the graph down by 5 units and then compressing it horizontally by a factor of 1/3.

4. The following table shows corn production in th US from 1986 to 1992. Compute the ARC in corn production between 1986 and 1990. Explain what this means in a sentence.

   Year	1986	1987	1988	1989	1990	1991	1992
   mill.lbs	1217	1387	1102	965	890	812	724

5. The initial amount of a certain drug in the body is 12 mg when you first take it. The amount decreases by 27% as each hour passes. How much of this drug will be left in the body after 2.5 hours?

6. Ally decided to run a lemonade stand in her neighborhood. The materials for building the stand cost $26.10. The lemonade only costs $0.10 to make, however, she sells it for $0.50.

   a. What are Ally’s fixed costs? What are the variable costs?

   b. Write and equation for the total cost as a function of the quantity.

   c. What is her revenue?

   d. Write an equation for the profit.

7. A company sells toasters. The demand curve is given by \( p=90-q^2 \) and the supply curve is given by \( p=q^2+4. \) What is the producer surplus when the price is at equilibrium?

8. Capital One’s stock price was $50.82 on September 12th and it rose to $63.36 on Sept 15th. Calculate how much the stock price increased or decreased between September 12th and September 15th.

9. A pair of jeans you are dying to get costs $108.99. A week late you return to the store and the jeans are now $78.99. How much did the store discount the jeans for?

10. The value of a Honda Civic is approximated by \( V=f(a)=13.78-0.8a. \)

    a. What is the significance of \( f(0)? \)

    b. For what value of \( x \) is \( f(x)=0? \) What is the significance of this value?

11. A company finds that there is a linear relationship between the amount of money that it spends on advertising and the number of units it sells. If it spends no money on advertising, it sells 400 units.For each additional $4000 spent, an additional 40 units are sold.

    a. If \( x \) is the amount of money that the company spends on advertising, find the formula \( Y, \) the number of units sold as a function of \( x. \)

    b. How many units does the firm sell if it spends $24,000 on advertising?

    c. How much advertising money must be spent to sell 1200 units?

12. The new boxing gym costs $500 to maintain and they pay their trainer $100 per month. A monthly membership is $150. Write a function that computes the profit of the company per month.

13. An amusement park has a fixed cost of $4000 per day and variable costs averaging $15 per customer. The park charges $20 per ticket. How any customers per day does the theater need to make a profit?

14. Solve the following equations:
    \begin{align} 7.586 a^6 &= 12.734 \\ a^5 &= 32 \\ 4.453 a^{18} &= 5.937 \end{align}

15. The average rate of change in profit for a company from 2002 to 2005 is 10.2 million dollars per year. Using this information, find the company’s profit from 2002 to 2005.

16. You deposit $5000 in the bank and the bank offers you 3% interest. How much is in the bank after 18 years if you compound it annually? Continuosly?

17. You have recently won the million-dollar lottery, and have 2 choices of receiving your funds. You can either choose to receive $920K once after having it sit in a bank for 3 years with 6% interest, or having $250k added to your account every year for 4 years with 6% interest added after the first year. Which would you pick? Justify your answer.

18. Scientists in Japan take samples of the nuclear radioactivity after one of the nuclear bombs were dropped in Hiroshima. The radioactivity was tested at 8.3 millirems per hour. The level of radiation decays continuously at a rate of \( k=-0.13. \)

    a. What was the level of radiation one week later?

    b. Find the number of hours until the level of radiation reached an acceptable limit of 2.34 millirems per hour.

19. Suppose that a pancake shop has a machine that costs $2,000 and is sold ten years later for $350. Assuming linear depreciation, how many dollars per year does it depreciate in value?

20. Columbia, South Carolina’s population was 119,370 people in the year 2000 and is growing by 900 people a year.

    a. Give a formula for our city’s population, P, as a function of the number of years, \( t, \) since 2000.

    b. What will the population be in 2025?

    c. When will the population reach 200,000 people?

21. Which of the following functions are power functions? State the constant of proportionality and the power.
    \begin{align} &y = 7 \sqrt[3]{x} && y = \big( 10x^4 \big)^3 && 11^{2x} \end{align}
22. What are the degrees and leading coefficients of the following polynomials?
    \begin{align} y = 2x^3+7x^2-10, && y = \tfrac{1}{4} + \tfrac{2}{3}x^9 - \tfrac{5}{7} x^5, && y=6 \end{align}

Derivatives. Section 003

Many typos, incomplete questions, questions with missing information, repeated topics among different students, and problems that belonged in another group. Not good. I had little choice among the surviving problems. 2/5

Derivatives. Section 019

A better choice, without typos, or different students sharing a same topic. 5/5

1. Let \( f(x) = 5^{3x}. \) Use a small interval \( (4 \leq x \leq 4.001) \) to estimate \( f’(4). \)

2. Compute the instantaneous rate of change for the function \( P(t) = 30(1.7)^t \) at \( t=4. \)

3. Compute the derivative of the following functions:

   \begin{align} &f(x) = 7e^{7x^2} && f(x) = 14^{3x^2+2x-4} && f(x) = \sqrt{ x^6 - \ln (5x^2) } \\ &f(x) = \frac{3500\pi}{2} && f(x) = 45x+3 && f(x) = (37x-\pi)(1+2x) \\ &f(x) = x^4 + x^6 && 6x^5 +\frac{3}{x^2}+\sqrt[4]{x} + \frac{1}{\sqrt[4]{x}} && f(x) = e^{2x} \\ & f(x) = 4^x && f(x) = \ln (3-x^5+2^x) && f(x) (5x^4-15)(3^x+2^x) \\ &f(x) = \frac{3x^2+\ln(x)}{1-3x^{4/3}+2^x} && f(x) = \frac{x+2\pi-\ln \pi + \pi^x}{e-e^x+3x^2} && f(x) = \frac{x^8+2}{x} \\ & f(x) = 8^{2x} x^2 && f(x) = \big(7+ \ln(x^3+14) \big)^{0.9} && f(x) = 5e^{10x} + e^{-x^3+6x^2-4} \end{align}

Applications of Derivatives. Section 003

Many typos and incomplete questions, as well. The selection of problems is very good, though, and you have chosen pretty much all the required skills. Some questions were actually fun. 4/5

Applications of Derivatives. Section 019

This is a decent selection of problems, but I am missing some key exercises. 4/5 as well.

1. The cost (in dollars) of producing \( q \) iPhones is given by \( C(q) = 17q^{10} +1981q+526912. \) Find the marginal cost function. Compute \( C(200), MC(200), \) and explain what it means.

2. Assume that \( C(q) \) and \( R(q) \) represent the cost and revenue (respectively) in dollars, of producing \( q \) items.

   a. If \( R(65) = 5400 \) and \( MR(65) = 70, \) estimate \( R(68). \)

   b. If \( MC(65) = 45 \) and \( MR(65) = 70, \) approximately how much profit is earned by the 66th item?

   c. If \( C’(200) = 60 \) and \( R’(200) = 53, \) should the company produce the 201st item? Why?

3. The total revenue (in dollars) and total cost functions are given below. Find the quantity that should be produced in order to maximize profit with respect to \( 0 \leq 𝑞 \leq 4000.\)
   \begin{align} &R(q) = 100q - 0.2q^2 && C(q) = 300 + 2.4q \end{align}

4. Find the critical points of the following functions, and classify them as local maxima, local minima, or neither.

   \begin{align} &f(x) = 7x^3+2x^2-9x-400 &&f(x) = \frac{x^5}{5} - \frac{5}{3}x^3 + 4x -100 \end{align}
5. For the functions below, determine intervals of concavity, and locate all inflection points
   \begin{align} &f(x) = x^2-8x+4 &&f(x) = 4e^{-x^2} &&f(x) = \ln (1+x^2) \end{align}

6. Find the tangent line to the graph of \( f(x)=15-2x^2 \) at \( x = 1. \)

7. Find \( a \) and \( b \) so that the line \( y = 2 \) is tangent to the graph of \( y = ax^2 + bx \) at \( x=1. \)

8. For what values of \( a \) and \( b \) does \( f(x) = a( x - b\ln x ) \) have a local extremum at the point \( (4,6)? \)

9. Use the first derivative test to find all the critical points of the function \( f(x) = 3x^4 - 4x^3 - 12x +3,\) then use the sign chart to determine whether they are a local maxima or local minima

10. Compute the relative rate of change of the following functions:

    \begin{align} &f(z) = 3z^2 - 4 &&f(x) = \sqrt{x} + 54e^{3x} \\ &f(x) = \ln (9x-2) && f(x) = 3e^{2x} \end{align}

11. For time \( 0 \leq t \leq 50 \) in days, the rate at which a factory releases pollution is represented by the rate at which CO2 is emitted is given by \( P(t) = 200 \big( e^{-0.4t} - e^{-0.2t} \big). \) When is CO2 being released the fastest?

12. If a theatre of 2500 seats is sold out when tickets costs $10 and decreases by 25 for every $1 increase in price, what is the price per ticket that maximizes revenue?

13. Find the quantity that maximizes profits if the total revenue and total cost function are given by

    \begin{align} &R(q) = 14q - 3q^2 && C(q) = 240 + 5q, \end{align}

    Where q is the quantity and \( 0<q<10. \)

14. Let \( C(q) \) represent the cost and \( R(q) \) represent the revenue, in dollars, of production of \( q \) items.

    a. If \( C(50)=4300 \) and \( C’(50)=24 \), estimate \( C(52) \).

    b. If \( C’(50)=24 \) and \( R’(50)=35 \), approximately how much profit is earned by the 51st item?

    c. If \( C’(100)=38 \) and \( R’(100)=35 \), should the company produce de 101st item? Why or why not?

15. The function \( f(x) = x^3 - 2x^2 + 4x \) has a critical point at \( x = 2.) Use the second derivative test to identify it as a local maximum, local minumum, or neither.

16. Find the second derivative of the following functions:

    \begin{align} &f(t) = 3t^2 + 8t^{1/2} + e^t && f(x) = \ln \frac{8x^2}{11x^3+14} \end{align}

17. The amount of caffeine measured in mg in the bloodstream at time \( t \) minutes is represented by the function \( Q(t) = 20 (0.3)^t. \) Estimate the rate of change of the amount of caffeine when time equals 5 minutes. Interpret your answer.

18. Annual coal production in the US, \( W = f(t) \) in millions of tons is a function of \( t \) years since the start of 1990.

    a. Interpret the statements \( W(10) = 117, W’(10) = 8. \)

    b. Calculate the relative rate of change of \( W \) at \( t = 10, \) in terms of coal production.

Antiderivatives. Section 003

Decent, straightforward selection of integrals, with different difficulty levels. I liked it. Two students managed to write integrals that didn't make sense, or integrals that need to be solved with techniques that are not taught in MATH 122. Those students will not get any credit, instead of lowering everybody's score for this part. A solid 5/5

Antiderivatives. Section 019

The selection of this group was dissapointing. Very basic antiderivatives, without any real variation. Not representative of the depth and dificulty of the problems we have done in class. 2/5.

1. Find the following antiderivatives:

\begin{align} &\int x^5+x^{100}\, dx &&\int \frac{3}{x^{24}} + 3x^{-4} - 27x^3\, dx &&\int e^x - \frac{24}{x^{3/4}}\, dx \\ &\int \frac{x^2}{x^3+1}\, dx &&\int (x+1)\big( \tfrac{x^2}{2}+x \big)\, dx &&\int \frac{x-10}{x}\, dx \\ &\int 6x^2 e^{x^3}\, dx &&\int 4^{3x}\, dx &&\int (3x-2) (3x^2 -4x +5)^{26}\, dx \end{align}

Antiderivatives. Sections 003 and 019

Both sections have done a great job at captuting the different skills and difficulty levels of this part of the course. I had to pick a selection of all the problems, to avoid repetition. If your particular problem has not been featured here, do not worry. You still got a 5/5.

1. With \( t \) years since 2000, the population, \( P \), of the world in billions can be modeled by \( P=6.1e^{.012t} \)
   • What was the population in 2013?
   • What will be the population in 2025?
   • Find the average population of the world between 2000 and 2011.
2. If \( t \) is time in years, and \( t=0 \) represents January 1, 2005, worldwide energy consumption, \( r \), in quadrillion (\( 10^{15}\)) BTUs per year, is modeled by, \( r = 461e^{.0185t} \).
   • Write a definite integral for the total energy use between the start of 2005 and the start of 2017.
   • Use the Fundamental Theorem of Calculus to evaluate the integral. Round to the nearest integer.

3. Compute the following definite integral:

   \begin{equation} \int_1^4 \big( 3^x + \tfrac{3}{x} \big)\, dx \end{equation}

4. War Emblem, winner of the 2002 Kentucky Derby, can reach up to the speed of 50mph. His speed was clocked every 30 seconds. The results are in the table below. What was his average speed?

   Time (Seconds)	0	30	60	90	120
   Speed (mph)	0	40	38	35	37

5. The following table gives the number of likes, \( L \), that Amanda gets on her instagram post per hour. Let \( t \) be the number of hours since the post and \( L=f(t) \). What is the average number of likes that Amanda got in the first 6 hours?

   0	1	2	3	4	5	6
   0	23	30	37	32	13	70

6. Find the signed area of the function \( y= f(x) = -2 + 5x \) between \( x=0 \) and \( x=4 \)

7. The following table gives the emissions \( E \) of nitrogen oxide in millions of metric tons per year in the US Let \( t \) be the number of years since 1970 and \( E = f(t) \).

   Year	1970	1975	1980	1985	1990	1995	2000
   E	26.9	26.4	27.1	25.8	25.5	25.9	22.6

   • What are the units and meaning of \( \int_0^{30} f(t)\, dt \)?
   • Estimate the previous integral.

8. Find the consumer surplus for the demand curve \( p = 100 - q^2/2 \) when \( q = 4 \)

9. For the function given below, compute left and right Riemann sums, and average them.

   t	0	2	4	8	12	15	16
   f(t)	1	4	1	2	4	5	6