# Interesting calculus problems of medium difficulty?

I would like to know sources, and examples of good "challenge" problems for students who have studied pre-calculus and some calculus. (differentiation and the very basics of integration.) Topics could be related to things such as:

1. Taylor Series.
2. Product Rule, Quotient Rule, Chain Rule.
3. Simple limits.
4. Delta Epsilon Proofs.
5. Induction proofs for the sum of the first n, integer, squares, etc.
6. Integration by substitution.
7. Other topics...

What I have found so far are too many problems that are just a bit too difficult. The problem can have a "trick" but it needs to be something a freshman could do.

Here is one problem that I thought was just at the right level:

If $f(x) = \frac{x}{x+\frac{x}{x+ \frac{x}{x+ \vdots}}}$, find $f'(x)$*

*To be honest this problem makes me a little nervous. Still, I like it.

If $C_0 + C_1/2 + \ldots + C_n/(n+1) = 0$, where each $C_i \in \mathbb{R}$, then prove that the equation $C_0 + C_1x + \ldots + C_nx^n = 0$ has at least one real root between 0 and 1. (Rudin, Ch 5 Exercise 4)

If $|f(x)| \leq |x|^2$ for all $x \in \mathbb{R}$, then prove that $f$ is differentiable at $x = 0$. (Spivak?) (OK, so this one is easier than "medium" perhaps, but I like it anyway. It illustrates nicely that growth conditions can have an impact on smoothness.)

• +1 for your second problem in particular. Interesting and very important! – BBischof Nov 12 '10 at 6:57

The Missouri State problems are often very good

• I may be biased since I go there, but I like to think Dr. Reid's problems are pretty awesome. He actually puts them on little slips of paper in the hallway near his office for those of us who enjoy working with them in the department. – Tristen Aug 18 '13 at 3:28

Since they know "basic" integration:

Find the integer part of

$$\sum_{n=1}^{10000} \dfrac{1}{\sqrt{n}}$$

• I like this one :) – BBischof Nov 12 '10 at 7:12

I like this problem, and my students seem to like it also. Technically, they should use density of the rationals, but you can skirt around this.(At least the solution I am thinking of).

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x)+f(y)$.

and related:

Show a function is linear iff $f(x)=f(x-a)+f(x-b)-f(x+a+b)$.

• A condition like "for all $a$ and $b$" is required in the second problem. – T.. Nov 12 '10 at 7:14
• T is exactly correct. – BBischof Nov 12 '10 at 14:24
• The second equation does not seem to be satisfied by any linear function (other than constant functions). Can you please clarify? – Srivatsan Nov 17 '11 at 5:46
• @Srivatsan did you consider it for f(x)=x? – BBischof Nov 25 '11 at 6:23

Another one that is kinda silly, but a little fun is

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that;

$\int_0^{\pi/4}f\left(x\right)dx=\int_0^{\pi/8}f\left(x\right)dx=\int_0^{\pi/16}f\left(x\right)dx=\int_0^{\pi/32}f\left(x\right)dx=\dots$

Show that $f\left(\sqrt{t}e^t\right)=-\left(3t\right)f\left(\sqrt{t}e^t\right)$ for infinitely many $t$.

I recommend the following sources (big advantage: free access):

1) Mathematical Reflections: http://awesomemath.org/mathematical-reflections/

"Through the problem column, we challenge students to develop their creative problem solving and reasoning skills by devising solutions to the proposed questions. Exceptional solutions will be published, with the intent of encouraging students to formally write out and submit their work to be showcased in print".

2) School Science and Mathematics: http://ssmj.tamu.edu/problems.php

"This section offers readers an opportunity to exchange interesting mathematical problems and solutions. Proposals are always welcomed... The editors encourage undergraduate and pre-college students to submit solutions. Teachers can help by assisting their students in submitting solutions. Student solutions should include the class and school name".

This is a classic one due to Arnold (A Mathematical Trivium):

$${1 \over 2\pi} \int_{0}^{2\pi} \sin^{100}(x) dx$$

They should also try to find a numeric value (to within one-percent error, say).

A great source of problems I have used in the past is, "Problems in calculus of one variable", by I A Maron.

Here are some interesting problems:

• $\displaystyle \int\limits_{0}^{\frac{\pi}{2}} \sin{x}^{\cos{x}} \Bigl[ \cos{x}\cot{x} - \log(\sin{x})^{\sin{x}}\Bigr] \ dx$

• $\displaystyle \int \tan{x} \cdot \tan{2x} \cdot \tan{3x} \ dx$ : Why i say this is interesting because, many people would attempt to solve it directly but one can use the fact that $\tan(3x)=\tan(x + 2x)$

• There are many resources available in the web. The Harvard - MIT Mathematics tournament problems should be challenging.

Let $m$ be an integer $\geq 2$. Define $u_0=e^{1/m}$, and $u_{n+1}=\frac{u_n}{u_n-I(u_n)}$ where $I$ denotes integral part. Prove that $I(u_n)=mn$ for $n \geq 1$.

• Continued fraction? – Aryabhata Nov 12 '10 at 19:30
• Maybe, but it can be done with Taylor series. – Plop Nov 12 '10 at 19:36
• Maybe not, was just guessing. – Aryabhata Nov 12 '10 at 19:38

Variational calculus is always fun (for example, derive from Newton's law the shape of a chain left hanging from its two ends, you end up with cosh) and is a nice introduction to analytical mechanics. Or pursuit problems (http://mathworld.wolfram.com/PursuitCurve.html) were a real challenge, but were quite enjoyable. However, those a more dealing with differential equations.

You got me started...

Assume $f\left(x\right)$ is a function ith a continuous derivative and $f\left(1\right)=0$. Show $\exists c\in\mathbb{R}$ such that $c^2f'\left(c\right)+2cf\left(c\right)=0.$

• Differentiability is sufficient. – Did Aug 9 '11 at 0:45

The best site I have come across when it comes to these kind of problems, is

Regards

• +1: Can you please be more specific, though? What sub-forums for instance? – Aryabhata Nov 12 '10 at 15:17

Ok last one for a while, I promise.

You probably all know this one as it is quite famous:

Let $f:\left[0,1\right]\rightarrow\left[0,1\right]$, then define a point of period n, for $n\in\mathbb{N}$ to be a point $x_1$ such that \begin{equation} f^n\left(x_1\right)=\underbrace{f\circ f\circ\dots\circ f}_{n-times}\left(x_1\right)=x_1 , \end{equation} and that for no $m$ less than $n$, $f^m\left(x_1\right)=x_1$. That is, $n$ is the smallest natural number such that $f^n\left(x_1\right)=x_1$. So a point, $x_1$, has period $2$ if $f\left(f\left(x_1\right)\right)=f\left(x_2\right)=x_1$ and $x_1\neq x_2$. Now define a fixed point of f to be a point $x_0$ such that $f\left(x_0\right)=x_0$; i.e. it has period one. Show that if $f$ has a point of period $2$, then it has a fixed point.

• Some condition on $f$ must be missing, otherwise consider the function $f$ defined by $f(0)=1$, $f(1)=0$ and $f(x)=x/2$ for every $x$ in $(0,1)$, which has $0$ and $1$ as points of period $2$ but no fixed point. – Did Aug 9 '11 at 0:41

Suppose $\displaystyle f:\mathbb{R} \to \mathbb{R}$ and $\displaystyle g:\mathbb{R} \to \mathbb{R}$ are differentiable functions such that

$$f'(x) = 3f(x) + 7g(x)$$

$$g'(x) = 2f(x) + 3g(x)$$

Find all such $f$ and $g$.

Hint: Consider $\displaystyle h(x) = 2g(x) - f(x)$.

Given the level they are at, I suppose this is the right level of difficulty you are looking at.

A little more advanced, they can differentiate again and construct a second order differential equation in $\displaystyle f$ (or $\displaystyle g$), which will work with arbitrary coefficients.

More advanced than that, they can consider a matrix differential equation.

Suppose $\displaystyle f:[0,1] \to \mathbb{R}$ is a function such that

1) $\displaystyle g(t) = \lim_{x \to t} f(x)$ exists $\displaystyle \forall t \in [0,1]$.

2) $\displaystyle f(x) + g(x) = 2x \ \ \forall x \in [0,1]$

Find all such $\displaystyle f$.

The trick is to realize that $\displaystyle g$ must be continuous (perhaps this is near the medium/difficult border).

For extra credit, they can try showing that 1) is enough to show that there is at least one point where $\displaystyle f$ is continuous. (See this: Is there a function with a removable discontinuity at every point?)

In fact it can be shown that if 1) is true, the set of discontinuities of $\displaystyle f$ is countable (See the same question above).

Evaluate $\displaystyle \lim_{x\to0}\frac{x^2\tan{\left(\sin{x^2}-2x\sin{x}\right)}}{\sin{x^2}}$.