Limits+assessments

I would consider changing the oscillating problem to have "explain" - my students just give the answer of DNE -and I am not sure if they know the real reason why.

- same test as above, but as a word document so that you can make changes KEY:

I used this as a review this year: Worked out solutions: **Number 17 is incorrect on my answer key. I copied the problem down and factored incorrectly. You should have** **(x+1)(x-2)** **-** **(x+3)(x-2) Which would give you a v.a. at x=-3 and a hole at x=2**



- There are a couple things not on this test because I cut and paste them from other sources. The table and graph for #3 are from the AP Exam prep book for the FDWK textbook (page 72). The table for #5 is from the 2007 AP Exam, question #3.

1981 AB #5 BC #2: Catherine Nguyen

Let f be a piecewise function defined by f(x) = {2x+1, for x< or = 2, and (x^2)/2 +k for x>2 a) For what value of k will f be continuous at x=2? Justify your answer b) Using the value of k found in part (a), determine whether f is differentiable at x=2. Use the definition of the derivative to justify your answer. c) Let k=4. Determine whether f is differentiable at x=2. Justify your answer.

Some thoughts from Andy Oats: You could do a question that requires algebraic manipulation such as factoring or multiplying by a conjugate (of numerator or denominator) in order to get some canceling. However, this type of question could easily be done as a multiple-choice question (non-calculator of course so students don't just use the Table function of their calculator). So, what about trying something along these lines. If you have talked about how we can answer infinity or negative infinity as an answer sometimes for a limit that does not exist, you could ask the students to explain why the limit as x approaches 1 of the function 1/(x-1)^2 has a limit of infinity but the limit as x approaches 1 of the function 1/(x-1)^3 does not exist. This should test to see if they understand the concept of limits from the right and left needing to agree. Likewise, you could ask a question like this: find the limit as x approaches 8 of f(x) where f(x) = abs(x-8)/(x-8). Then add the common AP Calculus phrase, "Justify your answer." Again, students would have to figure out that the limit as x approaches 8 from the right is 1, but the limit as x approaches 8 from the left is -1, so the limit does not exist. Lastly, you could ask a question something like this: If f(a) = b, does the limit as x approaches a of f(x) = b? Explain your answer. (Or you could reverse the question, if the limit as x approaches a of f(x) = b, does f(a) = b?) Here students would have to think through the concept of a limit having a different answer than the actual value - usually where you have a function with a hole where the x-value of the hole is then defined somewhere else above or below the graph of the main function. Basically, I try to get some types of questions that gets the students to explain their understanding of limits as it deals with vertical asymptotes, right and left sided behavior, holes, etc.