# Write calculus multiple choice questions with only numerical answer choices?

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

• "Then $c + d$ equals ..." --- I realize this is probably an example you made up on the fly, but it shows how questions like this need to be looked over very carefully, ideally the 2nd or 3rd look being at some later time (hours, or days) after you first wrote and proofed the question. Thus, in this case you'll want to say that $\frac{c}{d}$ is reduced to lowest terms (with $d > 0$ if the value is negative), or include something like "which of the following could be the value of $c+d$ (which requires even more care to exclude other possibilities, plus it could test extraneous skills). Oct 20 '21 at 18:08
• Out of curiosity, what is the multiple-choice, computerized system you may be using? Does it not come with any questions pre-built? Oct 20 '21 at 21:24
• @NickC It's a prototype/beta system from a student project in another department. It comes with nothing so far. It's an empty database to start. I'll be the first. Oct 20 '21 at 23:16
• Why exactly is it a concern that non-numerical answers 'give away too much'? Too much of what? Oct 21 '21 at 8:41

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

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). 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 $$[2, 3]$$.

• This is a very nice answer! I hope the OP does not try to reinvent the wheel. There are many, many good multiple choice questions available from earlier AP calculus exams. Oct 20 '21 at 19:12
• @user52817 Where are these older AP problems? I was planning on using material from past exams from the past 10 years that I have. They are all free response problems, but I think I could modify most of them for numerical answers. It is very, very time consuming for me to create non-numerical answer choices but if it is a generally accepted better choice, then I will over time. Oct 20 '21 at 19:52
• I like this! I had forgotten that this style of choice ("I and II only") also appears on some actuarial exams. And on those exams I've never felt like the answers revealed too much. Granted, they are not that deep or conceptual, but still, I've enjoyed them. Oct 20 '21 at 19:58
• Note that those are not multiple-choice questions, but single-choice questions, some of which reduce a potential multiple-choice problem to single-choice in a convoluted manner. Oct 21 '21 at 11:58
• @Wrzlprmft No: the term "multiple-choice" refers only to the options provided; it does not require that more than one option can be selected.
– m90
Oct 23 '21 at 14:00