Write calculus multiple choice questions with only numerical answer choices?

Question

What are the best type of answer choices for multiple choice questions? Numerical only like the probability and financial math actuarial exams?

I have an opportunity to utilize a multiple choice computerized system to let students practice and prepare for exams for a first calculus course at a university in the USA. This opportunity would be an optional self-learning/self-assessment supplement; not a requirement.

Course grades are based on three in-person, hand-written, free-response exams and on weekly quizzes. Students are not allowed calculators or notes for them. They are to use analytic-type reasoning and are awarded partial credit. HW is never collected.

Since the only option is multiple choice I need to decide if the answer choices should be all only numerical or not. My concern is that revealing non-numerical answer choices gives away too much. But then numerical only options limits what types of questions I can ask I think.

I mean suppose I want to ask about the MVT. Numerically, I could ask to find a value in the interval guaranteed by the theorem. But I also think it's important for them to verify that the theorem even applies which seems to me to be non-numerical.

Example of a numerical question that does not reveal too much in my opinion:

Q. Suppose there exists whole numbers $c$, $d$, and $e$ in the interval $[1,9]$ such that $f(x) = c\sin(x)$ and $f^{\prime}(\frac{d}{e}\pi) = 4$. Then $c + d$ equals

A. 5 .... B. 6 .... C. 7 .... D. 8 .... E. 9

Answer 1

I think it is good to pose both numerical and non-numerical problems. While the numerical problems are generally the focus in a Calculus class, a strong conceptual understanding helps make everything easier. You specifically mention theorems, so you could have problems like the following:

1) Which of the following conditions are required to use the Mean Value Theorem?

I: $f$ is continuous on $[a, b]$

II: $f$ is continuous on $(a, b)$

III: $f$ is differentiable on $(a, b)$

IV: $f(a) = f(b)$

Answer Choices:

a) I and III only.

b) II and III only.

c) I, III, and IV only.

d) II, III, and IV only.

2) Which of the following is true for the function $f(x)=|x|$?

a) There exists some $c\in(-1, 1)$ such that $f'(c)=\frac{|1|-|-1|}{1-(-1)}.$

b) The Intermediate Value Theorem applies on the interval $(-1, 1)$.

c) All of the above.

d) None of the above.

3) Suppose $f$ is differentiable on $[a, b]$ and $f(a)<f(b)$. Which of the following is true?

I: You can apply the Mean Value Theorem on the interval $[a, b]$.

II: You can apply the Intermediate Value Theorem on the interval $[a, b]$.

III: You can apply Rolle's Theorem on the interval $[a, b]$.

IV: You can apply the Extreme Value Theorem on the interval $[a, b]$.

a) I and IV only.

b) II and IV only.

c) I, III, and IV only.

d) I, II, and IV only.

e) II only.

4) The graph of the function $f$ is displayed below. Which of the following is true about $f$? (Please pretend I added a graph here.)

I: You can apply the Mean Value Theorem on the interval $[0, 1]$.

II: You can apply the Intermediate Value Theorem on the interval $[0, 1]$.

III: $f$ is differentiable on $(0, 1)$.

IV: $f$ is continuous on $(0, 1)$.

a) III only

b) IV only

c) II and III only

d) I, II, III, and IV

5) Suppose $f(x)=x^2$. Find the value of $c$ such that $f'(c)=\frac{9-4}{3-2}$.

a) 2.5

b) 6.25

c) No such $c$ exists because $f$ is not continuous on $[2, 3]$.

d) No such $c$ exists because $f$ is not differentiable on $[2, 3]$.