Good final exam bonus problem for calculus students?

Question

I'm making a final exam for a first course in calculus at a university. I need some suggestions for a good bonus problem. By "good" I mean an interesting problem that some of the brighter students could solve in about 15 minutes.

Hopefully it should involve limits, derivatives, or integrals.

The problem should NOT involve trig functions.

The problem should NOT be theoretical. These students can't do real analysis proofs, for instance.

Answer 1

Here are a few interesting problems:

1. Let $f^n(x)$ denote the nth iterate of a function (for example, $f^2(x)=(f\circ f)(x)$). If $f$ has a fixed point at $x=x_0$ (meaning that $f(x_0)=x_0$) then what is $\frac{d}{dx}f^n(x_0)$ in terms of $f'(x)$?

2. Prove that the intersection of the parabola and the rectangle in the picture is $\frac{2}{3}$ the area of the rectangle:

3. A triangle starts is an equilateral triangle $ABC$ at $t=0$. However, each second, both $BC$ and the length of the altitude to $BC$ increase by $1$ unit. What does the measure of angle $BAC$ approach as $t$ approaches $\infty$?

Answer 2

Let $L_t$ be the line segment connecting $(t, 0)$ to $(0, 1-t)$, and let $R$ be the union of all these line segments for $t \in [0,1]$. Find the area of $R$.

(Inspired by a book I had as a child which claimed drawing these line segments would trace out a quarter circle -- which turns out to be decisively false. The result just looks vaguely similar to a quarter circle.)

Answer 3

1. Compute $$ \lim_{x\to0}\frac x{1-e^{x^2}}\int_0^xe^{y^2}\,dy $$

2. Let $p>1$; prove the following inequality $$ 2^{1-p}\le\frac{x^p+y^p}{(x+y)^p}\le1\;\;\forall x,y>0 $$

3. Let $f\in\mathcal C^2(\Bbb R)$ s.t. $$ |f(x)|<1\;\;\forall x\in\Bbb R\\ f(0)^2+f'(0)^2=4\;\;. $$ Show that there exists $\xi\in\Bbb R$ such that $f''(\xi)+f(\xi)=0$ (hint: consider the function $g(x):=f(x)^2+f'(x)^2$).

4. Let $\{a_n\}_n\subset[0,+\infty[$; then prove that $$ \sum_na_n\;\;\mbox{converges iff}\;\;\; \sum_n\frac{a_n}{1+a_n}\;\;\mbox{converges} $$

Answer 4

My suggestion:

1. Take any relatively complicated calculus problem.
2. Describe it entirely in words, with no diagrams, variable names, or math symbols.

And for a specific suggestion:

What is the volume of space enclosed by the outer limits of firing range of a cannon placed on a flat surface with gravitational pull equivalent to Earth's, free rotation in any direction, and a muzzle velocity of 400 meters per second, excluding any consideration of wind, air resistance, or bouncing shots?

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Writing the correct applicable equation given a specific verbal description of a problem is a key skill, and one that (in my opinion) quite well distinguishes the brighter students from those who can only solve a ready-made equation. Writing out the right equation is more than half the problem.

Another tricky one:

With a spaceship accelerating at a constant 1 Earth gravity for the entire 384400 km distance to the moon, reversing acceleration direction instantly at the halfway point, how long will it take to complete the journey?

Another, although this may require some trig:

An amusement ride centrifuge (in Earth gravity) with a radius (at the base) of 20 feet has its walls tilted outwards at 45 degrees. What speed (in revolutions per minute) must the centrifuge reach before objects inside will slide or roll up the walls?

(I think when I worked this myself I only needed basic geometry, but not 100% sure.)

Answer 5

This might be a tough one but whoever figures it out (without looking here...) is going to deserve the points:

Remind them that the derivative of a differentiable function is zero at a minimizer of that function, and that you can use this condition to locate the minimizer (e.g. finding the vertex of a quadratic).
Ask them to simply write down the above equation for the case where you're trying to find the curve $y(x)$ that has the minimum arc length from $x_1$ to $x_2$.
In other words, what expression do you need to take the derivative of, and with respect to what, in order to represent the fact that $y(x)$ has the minimum possible length from $x_1$ to $x_2$?

Clearly the curve would be a line segment, but that fact shouldn't really help them here. The idea is, if you didn't already know beforehand that such a curve would be linear, how would you start to prove it? Writing down the correct derivative and setting it equal to zero is the first step of that, which is what you're asking them to do.

Answer 6

"By considering the turning point(s) of the function $f(x) = x^{1 \over x}$ on $(0, \infty)$ prove that $e^{\pi} > \pi^e$."

Differentiating reveals a global maximum at $x = e$ implying $f(e) > f(\pi)$ and the result follows after raising each side to ${\pi}e$.

Answer 7

The following is a nice result of Euler:

Let $P(x)$ be a polynomial of degree $r<n$. Then $$ \sum_{k=0}^{n} (-1)^k \binom{n}{k} P(k) = 0. $$

There's a nice proof that simply applies $P(x \frac{d}{dx})$ to $(1-x)^{n}$. You can build up to it by asking for the proof for a monomial first, or start right at the beginning with $x\frac{d}{dx}$ acting on $x^k$, then calculating $x\frac{d}{dx}$ acting on the equality $(1-x)^n = \sum_{k=0}^{n} (-1)^k \binom{n}{k} x^k$, and so on. This was used on an entrance exam at my university a couple of years ago, with some guidance like this: the exact text of the question is as follows:

An operator $D$ is defined, for any function $f$, by $$ Df(x) = x\frac{df(x)}{dx}. $$ The notation $D^n$ means that $D$ is applied $n$ times; for example $$ D^2 f(x) = x\frac{d}{dx} \left( x \frac{df(x)}{dx} \right). $$ Show that for any constant $a$, $D^2x^a = a^2x^a$.

1. Show that if $P(x)$ is a polynomial of degree $r$ (where $r \geqslant 1$) then, for any positive integer $n$, $D^n P(x)$ is also a polynomial of degree $r$.
2. Show that if $n$ and $m$ are positive integers with $n<m$, then $D^n(1 − x)^m$ is divisible by $(1 − x)^{m−n}$.
3. Deduce that, if $m$ and $n$ are positive integers with $n<m$, then $$ \sum_{r=0}^m (-1)^r \binom{m}{r} r^n = 0. $$

So it does the monomial case. $P(D)$ would then be a generalisation of this notation: if $P(x) = \sum_{k=0}^{n} a_k x^k$, then $P(D) = \sum_{k=0}^{n} a_k D^k$; this is an easy extension of the monomial case since $D$ is linear.

Answer 8

I suggest this terse (Euclidean) geometry problem, which can be solved using basic calculus

$A$, $B$, $C$, $D$ are distinct points, with $A$, $B$, $C$ collinear,
such that $|AB|=|BD|=|CD|=1$ and $|AC|=|AD|$.

Give the full set of possible $|AC|$ values.

Solution (hover mouse to reveal spoiler):

$\left\{\,{\sqrt 5-1\over2}\,,\,{\sqrt 5+1\over2}\,\right\}$

Illustration (spoiler).

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As pointed in that other answer, writing the correct applicable equation given a specific verbal description of a problem is a key skill, thus alternatively the question might be

Alice, Bob and Carl stand on a straight line.

Alice is one furlong (220 yards) away from Bob.

Dana stands one furlong away from both Bob and Carl.

Carl is as far from Alice as Alice is from Dana.

How far can Alice be from Carl?

Answer using a well-formed sentence, with distance in furlong (exact, or with at least three significant digits).

Answer 9

Here was one I gave to my students in first-semester calculus as a bonus problem on a regular midterm (not a final exam). It's taken from one of the usual college textbooks on calculus. You can always modify it a bit. A figure goes with it.

Define $F$ by $F(x) = \int_{0}^{x} f(t) \, dt$, where the graph of $f$ is shown.

1. Compute: $F(0)$, $F(3)$, $F(6)$.
2. State the critical numbers of $F$ that are in the open interval $(0, 7)$.
3. For each critical number found in the previous part, determine if $F$ attains a local maximum value, a local minimum value, or neither at that critical number, and explain why.

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By the way, you should check out some of the past exams given out at, say, MIT and Stanford. Some of the problems are very good, having many parts (part a, part b, etc.) and involving many key topics.

Answer 10

Browse old problems from the Putnam exam. http://kskedlaya.org/putnam-archive/