Calculus AB · Unit 1 · Paper 2

Limits & Continuity unit test

A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.

Paper 1 Paper 2 Paper 3

Each paper is built from this unit’s 41 terms and is the same for everyone, so a teacher can assign “Unit 1, Paper 2” and every student sits the identical test. Multiple choice is marked objectively; the written sections you mark yourself against the model answer and rubric.

Suggested time 36 min 33 points0/17 attempted

A table cannot prove a limit

Canceling is valid only for x ≠ a

Continuity of piecewise functions

Vertical asymptote

Limits at infinity of rational functions

Infinite discontinuity

Three reasons a limit fails

Limits involving absolute value

End behavior by degree comparison

IVT requires continuity on a closed interval

Right-hand limit notation

Showing a limit does not exist

Short answer 1. Define or explain: Choosing k for continuity

3 pts

Short answer 2. Define or explain: Left-hand limit notation

3 pts

Short answer 3. Define or explain: Squeeze theorem structure

3 pts

Short answer 4. Define or explain: One-sided limits

3 pts

Free response

9 pts

NO CALCULATOR. The function f is twice differentiable. The table gives values of f and its derivative f′ at selected values of x. x 0 2 3 6 f(x) −1 3 8 5 f′(x) −5 4 9 −2

A. Find lim(x→2) f(x)/x, or state that the limit does not exist.

B. Let g(x) = f(f(x)). Find g′(2). Show the work that leads to your answer.

C. Let h be a differentiable function such that h(0) = 10 and h′(x) = f′(3x). Find h(2). Show the work that leads to your answer.

D(i). Let k be the function defined by k(x) = ∫₀ˣ t²·f(t) dt. Find k′(x).

D(ii). Find k″(3). Show the work that leads to your answer.