Quiz Solutions

Everyone in this class should be able to do all the problems on this assessment. I was discoraged to see that no one who took the assessment got 100%. A lot of people had no idea what to do in problem 3 where there was an x in the denominator. A lot of people had trouble solving for x when it is in the denominator as in problems 4 and 5. Lastly, (x+1)^2=x^2 + 2x +1 NOT x^2 + 1. Make sure you can do ALL the problems, especially the ones you missed. Don't hesitate to ask me questions if you have them. If you missed more that two problems I highly recomment going to the UT Learning Center's Algebra Review TWH 9/2 - 9/4.

Problem 1 was worth 3 points (one for each fill-in-the-blank), probelme2 was worth 2 points (one for each fill-in-the-blank),problem 3 was worth 5 points (one point for a,b,c; 2 points for d), and problem 4 was extra. The mean was about 6.4. I think you guys did okay overall. For the first fill in the blank on problem 1, I gave full credit for words like: separate, distinct, different. For the third fill-in-the-blank on problem 1 I gave full credit for words like: x-intercept, y-intercept, intercepts, points. My main comment for problem 3 is to make sure you show and label your work. Your goal should be to convince me that you know what you are doing, do not just assume that I know.

Problem 1 was worth 4 points, probelme2 was worth 1 point, problem 3 was worth 7 points (one point for each of a,d,e; two points for each of b,c). The mean was about 6.8 out of 12. I was pretty happy with the responses for the first probelem. I was disapointed with the responses for problem 2. Even after mentioning exactly what to look for in the reading, very few people got this correct. I did not give any partial credit for this problem. I have two comments for problem 3a. One, the range include 3 so you need to use a square bracket there, and, two, when you are writing an interval the samller value appears on the left. For problem 3b, a lot of people incorrectly wrote (-\infty,3) and (3,\infty) for the intervals on which f is increasing and decreasing. Remember that for this interval, you are looking at the values in the domain (i.e. input values), not value in the range and not points on the graph. Most people got most of problem 3c. It is fine to say that a function is even because its graph is symetric with respect to the y-axis. For those of you who wrote f is even because f(x)=f(-x), you should back this up by showing it is actually a true statement for ALL real values in the domain. A lot of people got problems 3d and 3e incorrect. It is important to understand the difference between the real number at which a function, f, has a local maximum and a maximum value of f. The former is the Real NUMBER IN THE DOMAIN of f whose value is a maximum (i.e the value you have to put into f). The latter is the real number f takes on or achieves (i.e the falue you get out).

This was a "fake" quiz, i.e it does not count for anything towards your grade. The purpose of this quiz was to help prepare you for the upcoming midterm.

Problem 1 was worth 1 point and problem 2 was worth 9 points (each part was worth 1 point except for the part a which was worth 2 points). The quiz looked pretty good in general. My main comment is to make sure you read the problems carefully. Prolem 2a asked you to write a function in a certain form BY COMPLETING THE SQUARE. Very few people actually completed the square and so very few people received much credit on this part of the quiz.

Problem 1 was worth 1 point and problem 2 was worth 9 points (each part was worth 1 point except for the part f which was worth 2 points). Make sure you know the difference between the degree of a polynomial and the multiplicities of the zeros. When asked for the behavior of a function (usually near a zero or for very large or very negative input values), you are being asked for an algebraic expression (not a wordy description). Also, when describing the behavior of a function g, near a zero for example, it is not mathematically sound to write g=(blah). It is better to write g~(blah). The"~" symbol means "approximately".

Problems 1 and 2 were worth 1 point each. Problem 3 was worth 3 points and problem 4 was worth 5 points (1 point for each part). On prolem 3, a lot of people wrote the inequatlity as x^2(x^2 + 3) \ge 0 and then proceeded to say that the zeros of the LHS were 0 and \pm\sqrt(3). However, x^2+3 =0 has no real solutions and the only zero of the LHS is 0. The only other comment I have about problem 3 is that people need to better label their work. Just writing down an interval and the function evaluated at a point is not sufficiet. I would suggested writing a few words such as, "since x^2(x^2 + 3) is positive for x=-2, the function x^2(x^2 + 3) is \ge 0 for all x in the interval (-\infty,0)." In Problem 4b a lot of people said that x=1 was a zero of the rational function, R(x). Now R(x) is not even definted at x=1 so how can it possibly be a zero? The zeros of a rational function which is in lowest terms are the zeros of the numerator only.

Problem 1 was worth 1 point, Problem 2 was worth 2 points. Problem 3 was worth 4 points. Problem 4 was worth 9 points (6 points for part a, and 1 point each for parts b,c and d). The second fill-in-the-blank on problem 2 was very similar to part of a problem that was not only on the second midterm but also discussed in class. You guys really need to know what a domain of a function is and how to use and equation to describe it. You will most likely see something similar to this on another exam. Problem 3 was just like one of your homework problems yet hardly anyone got it. Most people seemed to know that if the a^u=a^v then u=v but were unable to write the given equality so that both sides of the equation were exponents with the same base. One way to do this is to write 9 as 3^2. Problem 4a turned out to be harder than I wanted it to be. On part a, as long as the graphs you drew were labeled and matched the functions which labeled them you got full credit.

Problem 1 was worth 2 points. Problem 2 was worth 4 points. Problem 3 was worth 4 points. Problem 4 wasworth 5 points (2 points each for parts a and b and 1 point for part c). In problem 2, most people had the right idea (to write 2log(x+1) as log((x+1)^2) and then use the property that the difference of two log's is "log" of the quotient). However, a lot of people did also make a pretty bad mistake by writing that log(a)-log(b)=(log(a))/(log(b)). This is NOT true. log(a)-log(b)=log(a/b) which is NOT the same as log(a)/log(b). Problem 3 was taken, number for number, from your homework and hardly anyone got this problem correct. I even wrote the equation in a form suggestive of using the change in base formula. Using the change of base formula, one notices that x=log_3(9) = 2. For this problem some people use the change of base formula to write x=(log_3(8))(log_8(9))=(log(8)/log(3))*(log(9)/log(8))=log(9)/log(3). This is fine but still needs to be simplified. You can use the change of base formula again to write log(9)/log(3)=log_3(9)=2. I was happy to see that most people not only correctly figured out the domain of log_2(x+3) but wrote the equation x+3>0 to describe the domain. In problem 3b most people had trouble with the graph. A lot of people provided the graph of an exponential function rather than a logarithmic function. Also, people did not correctly shift the graph of log_2(x) correctly to get the graph of log_2(x+3). Some people labled their graph with points that are not on the graph of f(x). Finally, there were a handful of people whose graph had a domain of all negative real numbers even though in part a they had said that the domain was all real numbers greater than -3.

Problem 1 was worth 2 points. Problem 2 was worth 3 points. Problem 3 was worth 3 points. Problem 4 was worth 2 points. I was disapointed with the answers for problem 2. There was a lot of bad algebra and very few people seemed to have any idea how to do attack this problem. There is no reason for you guys not to know how to do this kind of problem. I went over this type of problem in class. You guys worked in groups to solve a problem just like this one. Lastly, this problem is again number-for-number from you homework. If you see something more than once in lecuture, on a quiz or on homework its probably a good thing to know how to do. If you have questions, please come ask in office hours rather than getting a low score on a quiz.

Problem 3 was again taken number-for-number from your homework. Most people know how to do problem 3b which is good. Some people had a bit of trouble with probelm 3a. A lot of you knew the angle was greater than 2\pi radian, which is good. However many of you did not know what quadrant it is in. One way to see this is to write 16(\pi)/3 as 4(\pi) + (\pi) * (\pi)/3. The first summand tells you that the angle goes around the origin two full turns. The second summand tells you that the angle goes another (\pi) radians and the last term tells you that the agle goes (\pi)/3 past (\pi) so that you end up in the 3rd quadrant. If this reasoning is something you have trouble with, it is a good thing to practice. I know problem 4 was probably a bit of a surprise since you did not have a homework problem on it but I did talk about and do an example with arclenght in class. If you did not know how to deal with arclength along a circle of radius different than one, you still should have known how to find the arclength along a circle of radius 1. Finding the arclength corresponding to an angle \alpha (radians) along of circle of radius 1 is synonymous with giving a definition of a radian and this you all should certainly know.

Problem 1 was worth 3 points. Problem 2 was worth 8 points. Problem 12 points. Overall you guys did very well. I am pleased.



Mideterm III Review (in lui of Quiz 13) (11/19/08) questions (page 1)

Here are some things you may want to study for Midterm III

Problem 1 was worth 20 points. Problem 2 was worth 11 points. Problem 3 was worth 4 points. Problem 1 was fairly good. In problem 1a, explaining how one can use a rod and its shadow to measure the angle of inclination of the sun involves more than just saying "tan(\theta)=opp/adj" (which a lot of you did). You do actually have to say which length is the opposite side of a triangle and which length is the adjacent side. It also seemed a lot of people did not really understand what the angle of inclination was. I got a lot of interesting responses for part 7. Most of you correctly indicated the angle, \theta, of inclination in your diagrams but then wrote that the angle of inclination was tan(\theta)="rod"/"shadow". While,tan(\theta)="rod"/"shadow" is a true statement, the angle of inclination is the acutal angle, \theta, and \theta = tan^(-1)"rod"/"shadow" . Lastly, for problem 1, you guys really need to watch your notation. It is very important that you both write true statements and write what you mean. A lot of people wrote things like "tan = a/b" or tan(\theta)=theta) or tan=a/b=30 degrees. None of the statements make sense. Make sure you understand why.

In problem 2 some of you had trouble with the units. AM and FM radio stations are measure in KHz (kilahertz) and MHz(megahertz) not just hertz. Also in computing the wavelength the frequency units you need use are Hz. This means you had to write khz as 10^3hz and mhz at 10^5hz. More importantly a lot of you had trouble with understanding the relationship between the period, p, and the scale factor w. The wavelength of your station is number of cm/cycle i.e. is the period (not the scale factor as several of you wrote down). The function y=sin(wx) has period p=2pi/w. Since you know the period (i.e the wavelength), you can solve for w. For example if suppose my wavelength is 300cm/cycle. Then my period p=300cm and I know 300=2pi/w. So I can solve for my scale factor w, w=2pi/300=pi/150. Since I now know the scale factor I know my electromagnetic wave can be described as y=sin((pi/150)x)

I was happy to see that just about everyone got problem 3 correct. I am glad that you guys understand that the statement sin^2(x)+cos^2(x)= 1 means that for every real number x, the sum sin^2(x)+cos^2(x) is always 1.

Problems 1 through 4 were each worth 1 point. Problem 5 was worth 2 points. Problem 6 was worth 4 points. The bonus problem was worth 1 point. In general you guys did pretty well. I was happy to see that most people knew how to approach problem 6. Most of you did problem 6 using the scaling method which is great. There was one main issue in problem 6. A lot of you forgot to check what quadrant the angle \theata was in. Knowing what quadrant the angle is in is very important for figuring out the sign of the other trigonometric function at the angle. Don't forget to figure out what quadrant the angle is in. Make sure you know how to do problem 5 when \sqrt(5pi)/8 is replaced by 5pi/8. A handful of people had trouble with the bonous problem. While a lot of you seemed to know that the period of the function f(x)= -tan(2x) is pi/2, a lot of you seemed to be confused at how to graph this. In the graph of a tangent function with period pi/2, whatever happens between -pi/2 and pi/2 in the graph of a tangent function now happens between -pi/4 and pi/4.