Jeopardy! Of calculus

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Two young mathematicians discuss a ‘Jeopardy!’ version of calculus.

Check out this dialogue between two calculus students (based on a true story):

Devyn

(Pretending to be Alex Trebek) I’ve got a new costume.

Riley

Whoa! You look just like Trebek!

Devyn

In Jeopardy!, I, Trebek, give you an answer, and you must tell me the question.

Riley

Uh Alex, ‘What are the rules of Jeopardy!?’

Devyn

Ha. Exactly! Let’s play a different version where I’ll tell you a derivative, and you tell me the function. Are you ready?

Riley

I’ll take “Formulas for slope” for $\$200$ .

Devyn

$3\cdot e^{3x}$

Riley

I’ve got an answer! Actually, I’ve got three different answers, I mean questions!

(a): “What’s the derivative of $e^{3x}$ ?
(b): “What’s the derivative of $e^{3x}+1$ ?
(c): “What’s the derivative of $e^{3x}-1$ ?

Devyn

Hmmm. Now I’m not sure which one it was.

Riley

What about if you had given me $\frac {\sin (x)}{x}$ ?

How many functions whose derivative is $3\cdot e^{3x}$ are there?

Zero One Two Three Four Infinitely many

How many functions whose derivative is $3\cdot e^{3x}$ that equal $1$ at $x=0$ are there?

Zero One Two Three Four Infinitely many