Published: Mar 26, 2024

Last Modified: Apr 10, 2024

This is a handout for a talk that I first gave on 2024-03-27 at David and Mary Thomson Collegiate in Scarborough.

This talk revisits the topic of Infinite Limits in Python, but without Python. It addresses a question that really vexxed me in highschool: “What does the $\sin(x)$ button on a calculator really do?” I remember asking my teaching this question and getting the response: “Oh – You’ll learn all about that in university.” This workshop is an attempt to answer the question in terms that I would have understood back in grade twelve.

The handout for this workshop is available here.

- Explain how the $\sin(x)$ button on a calculator calculates sine of $x$.
- Create “designer” polynomials.
- Learn about univeristy mathematics.

**Q1.** Get out a calculator and calculate $\sin(1.2)$ where $1.2$ is measured in radians.
(Note: If you get $\sin(1.2) \approx 0.0209$ then your calculator is in degrees mode.)

**Q2.** What do you think your calculator does when it calculates $\sin(x)$? How does it do it?

**Q3.** Consider the polynomial $\displaystyle p(x) = 2 + \frac{2}{1 \cdot 2}x^2 + \frac{4}{1 \cdot 2 \cdot 3}x^3$.
These fractions are left “uncancelled” and “unmultiplied” for a good reason: to help spot patterns.
Evaluate the following:

- $p(0)$
- $p^{\prime}(0)$
- $p^{\prime\prime}(0)$
- $p^{(3)}(0) \longleftarrow$ This $p^{(3)}(x)$ is the third derivative of $p(x)$.

**Q4.** If you wanted a polynomial $p(x)$ so that $p^{(n)}(0) = C$ and $p^{(k)}(0) = 0$ for $k \neq n$.
To put it another way: the $n$'th derivative of $p(x)$ is $C$ and all other derivatives are zero.
How would you build it? What is the formula for $p(x)$?

*Suggestion:* If this question is too abstract, pick your favourite numbers $n$ and $C$.

**Q5.** Consider the function $f(x) = \sin(x)$.
Calculate the first few derivatives $f^{(n)}(0)$.

- $f^{\prime}(x) = \quad \quad f^\prime(0) = \quad \quad$
- $f^{\prime\prime}(x) = \quad \quad f^{\prime\prime}(0) = \quad \quad$
- $f^{(3)}(x) = \quad \quad f^{(3)}(0) = \quad \quad$
- $f^{(4)}(x) = \quad \quad f^{(4)}(0) = \quad \quad$

**Q6.** Do you notice any pattern in the values of the $n$'th derivative of $\sin(x)$ at $x=0$?
Let $f(x) = \sin(x)$. If $n = 2k + 1$ is odd then $f^{(n)}(0) =$

And now we have all the pieces that we need for the Big Idea.
Suppose that two functions $f(x)$ and $p(x)$ have same value at $x=0$. That is: $f(0) = p(0)$.
Furthermore, suppose that their first derivatives agree at $x = 0$. As a formula: $f'(0) = p'(0)$.
Let's go really wild and suppose that $f^{(n)}(0) = p^{(n)}(0)$ for all $n$.
These functions will be *very* similar:
$p(x)$ will approximate $f(x)$.

**Q7.**
Let $f(x) = \sin(x)$ as above.
Use Q4 and Q6 to design a polynomial $p(x)$ of degree seven so that:
\[
f^{(n)}(0) = p^{(n)}(0) \textrm{ for all $n \leq 7$.}
\]
**Q8.**
Use your polynomial from Q7 to estimate $\sin(1.2)$ and compare it with your answer from Q1.

**Q9.** What *is* the calculator doing when you hit $\sin(x)$?

**Q10 (Further Exploration).**

- Use Desmos to plot a polynomial of degree $N$ that approximates $\sin(x)$.
You will want to type
`sum`

to get a summation sign $\displaystyle \sum_{n=0}$ and $n!$ for the factorial $n! = 1 \cdot 2 \cdot 3 \cdots (n-1)n$. - Can you do this polynomial approximation method for other functions? How about $\cos(x)$? Or $e^x$?

Thanks for reading! If you have any comments or questions about the content, please let me know. Anyone can contact me by email.