## Essay Questions in Beginning Calculus Classes – An Example

Occasionally I rant about separating  tests for skills from tests for understanding.  Here is an example of an essay question designed to see if a student knows the different methods of locating a local minimum value. Though this question is still less about concepts than it is about tactics, it will  check a student’s grasp of how to use derivatives and second derivatives.  I haven’t taught calculus in a while so this is all untried speculation.

Question 1

a) (10 points)  List the steps with brief explanations that you would use to show that $x=2.55$ exactly locates a local minimum of this function: $2x^3+3.18x^2-55.233x+6.84$? You may assume that you know all the calculus skills and concepts we have discussed since the beginning of the term, have all the algebra skills taught in high school and can perform arithmetic operations $(+,-,x,/)$ by hand to as much precision as necessary.  Note you are not working this problem, just telling how you would work it.  The assumption is that you do not have access to a calculator or any other (on-line) computer algebra system like Wolfram Alpha.  Consideration should be given to efficiency.  Fewer steps are usually better and fewer arithmetic operations are also usually better.

b) (5 points)  How would you answer the above question if in addition you had access to a graphing calculator like the T-83 we use in class but with only the graphing capability with zooming and tracing and only access to the zero and value functions in the CALC menu?  Note we still want to insure we have an exact answer.

c) (5 points) How would you use a computer algebra system like Wolfram Alpha to answer the problem in part a).  Specify the exact queries you would use.

Note:  This question has a lot of words.  If desired, the criteria for each section could have been reviewed with the class before the quiz, but still, additional time must be allocated for students to read the problem carefully.  Also non-native english speakers need to be considered.