Applied Calc Spring 2012 Exam 1 Results

Results for Exam 1

This webpage will give an overview of how the class performed as a whole on the exam. If you are dissatisfied with your performance on the exam, please make an appointment to see me. I may be able to suggest ways to make your preparations more effective for the remaining exams. And I am open to any suggestions about how I can make the course more supportive of your efforts.

You can find your individual score posted on Blackboard. When you get the exam back in class, double check that the score on your paper is the same as what is recorded on blackboard. Also, add up your points for the entire exam and double check that I have the right total for your paper. I am very careful totalling and recording scores, but mistakes CAN occur.

If you have not already done so, see a recent announcement on blackboard about assignments and other information about next week. 


Class Performance. The histogram below shows all of the scores for the first exam. The scores are rounded to the nearest whole number, so if you got an 86.5 that would appear as an 87 on the graph. The graph shows, for example, that two students scored 98 and one scored 95. There were 10 scores in the 90s, 12 in the 80s, 3 in the upper 70s and 1 below 70. At this point almost everyone is on track to earn an B or better in the course.  If you scored below 80 and are hoping to earn a B or better, you should probably be thinking about ways to improve. It might be a good ideas to come see me in my office in the next week or so to discuss the exam results.

histogram

Partial Credit Interpretation.  I try to give out partial credit on a percentage basis so that, for example, an answer that is indicative of B understanding will get around 80% of the points for that problem.  Even on a problem left blank you are unlikely to get 0 because that lowers your overall score so much.  On any problem where you got less than 70%, I am telling you that your answer indicates an unsatisfactory understanding of the material.  If your score is 50% or less of the available points, I am saying that your understanding of that particular item would not merit a passing grade.

Comments on Specific Problems.

Problem 3a. Only a few people remembered this idea, something I presented in class.  I tried to emphasize it but apparently did not make a big enough deal about it.  There are two somewhat different but closely related ways to think about what a derivative is.  In this item, I am asking you to recall the one that involves repeated magnification at a point on the graph of a function.  When you repeatedly zoom in, either the graph will appear to become a straight line, or it won't.  In the first case, if it does seem to become straight, then the slope of the the resulting line is the derivative.  Therefore, we can think of the derivative as the slope you see when you magnify until the curve straightens out.  On the other hand, if the curve never becomes straight, then the derivative does not exist at the point in question. 

The second way to think of a derivative, probably the more familiar, is that the derivative is the slope of the tangent line.

Problem 3b.  In this problem there were a large number of notation errors.  These are instances of students using terminology or mathematical symbols incorrectly.  That is important because it interferes with communication.  For example, I believe some students have been using the term "line" to mean any sort of trace you can draw with a pencil.  In mathematics though, line is always understood to mean straight line, and any other sort of trace is called a curve.  (Actually, a line can considered to be a curve in the same way that a square is considered to be a rectangle -- it is a very special case -- but it is more specific to call it a line.)  Anyway, if you say "the line" to indicate the graph of the function, and if I read "the line" and think you are talking about a straight line, you can see how confusing that would be.

The point of these remarks is to explain why it is important to use proper notation and terminology.  So for the case at hand, the curve that represents a function is given by an equation y = f (x).  That is, it is proper to refer to the curve y = f (x) but not to the curve f (x).  However, it is proper to refer to the curve as the graph of the function f.  A point on the curve or on the graph of f has to be a pair (x,y), not a number.  So, it is incorrect to say "at the point 2 on the curve".  You can say "at the point where x = 2" (there is only one possible point so described because f is a function and has just one point for each x).   In conclusion, a proper statement of the answer to b is "The notation f '(2) represents the slope of the tangent line to the curve y = f (x) at the point where x = 2. "

Problem 3c.  The average rate of change should be defined in terms of two points on the graph of a function, or between two values of x.  One way to do that would be to simply define the calculation: "The average rate of change of the function f(x) between x = a and x = b is defined to be the slope of the line from (a,f(a)) to (b,f(b))."  That is simple, direct, clear, unambiguous, and computable.  A more conceptual alternative might be to say that the average rate of change of a function between two points is the rate which, if held constant, would produce the same change in the function as is actually observed.  So for example, if you assume that the function is changing at a constant rate (equivalently that the graph is a straight line) between a and b, the average rate of change specifies what the constant rate must be to get from f (a) to f (b).   For a complete answer you would also have to say what instantaneous rate of change is and indicate how it is defined in terms of a limiting value of a process of successive approximation.

Problem 6a.  Here another terminology issue arose for several students who do not seem to recognize that "y intercept" has a very specific meaning.  At the risk of being overly repetitive, if you do not know the accepted meanings of definitions, notation, and terminology, you will not understand what I am asking on exam questions.