What do functions do for tiny x?
Looking at small values of functions. Motivating the existence of the Maclaurin expansion.
Problem
Archimedes, Bernoulli, Copernicus and de Moivre each try to think of a function which will be small near to the origin. Their choices are:
$$ A(x) = \sin(x) \quad B(x) = 1 - \cos(x) \quad C(x) = \ln(1+x) \quad D(x) = 1 - \frac{1}{1-x} $$
All of these functions equal zero when $x$ is exactly zero, but the friends want to investigate how small their functions are when $x$ is small but not exactly zero.
Use a spreadsheet or graph drawing software to investigate these curves for smaller and smaller values of $x$ (you may like to try graphs with $x$ ranging from -1 to 1, -0.1 to 0.1, -0.01 to 0.01 to begin with). Before you start, try to guess what will happen close to the origin in each case.
As you zoom in the scale what do you notice happening to the graphs in each case? What similarities do the functions have as we zoom in? What differences do they exhibit?
Can you predict the shapes of the graphs for x between -0.000001 and 0.000001? Test your predictions.
For each function can you write a simple polynomial approximation with 1 or 2 terms for the shape of each curve for very small values of x?
Getting Started
You may like to use Desmos, Geogebra or the example spreadsheet.
Visually what do you notice when you increase the magnification on the origin?
Student Solutions
Congratulations Andrei for another very good solution.
The 4 functions are: $$A(x) = \sin x,\quad B(x) = 1 - \cos x,\quad C(x) = \log (1+x), \quad D(x) = 1- {1\over (1-x)}.$$ I consider the logarithm in base e. First I plotted the 4 functions using Graphmatica. In all figures $A(x)$ is violet, $B(x)$ is white, $C(x)$ is red and $D(x)$ is cyan. For $[-1, 1]$ I obtain:
For [-0.1, 0.1]
And for [-0.01, 0.01] I have:
Each of the 4 functions could be approximated using the Taylor series expansion around 0, and the accuracy of the approximation becomes better for values of $x$ nearer to the origin. I shall use the second order polynomial: $$A(x) \approx x,\quad B(x) \approx 1- (1 - x^2/2) = x^2/2, \quad C(x)\approx x - x^2/2, \quad D(x) \approx - x - x^2.$$
Teachers' Resources
Why do this problem?
This problem begins to motivate the existence of the Maclaurin series in which any function which is well-behaved at the origin can be written as an 'infinite polynomial' or 'power series'. Students can begin approximating the coefficients of the polynomial numerically, before calculus is introduced in cases where the derivatives of the function needing to be approximated are known.
Possible approach
Have students predict the behaviour of the functions listed near the origin before drawing the graphs using Desmos or Geogebra. Make further refinements to these predictions before zooming in on the section of the graph around the origin.
This initial discussion can be done as a class with the teacher entering functions and zooming in, but it would be helpful for students to have access to their own graph drawing technology though phones, tablets, computers or graphics calculators for the next section. At this point they should try to approximate each function using a polynomial, investigating the most suitable coefficients for themselves. Working with a partner at this stage may help students to articulate their thinking, keep track of previous attempts and persevere in the face of difficulty.
Students may expect polynomials to be given in descending powers of $x$, but could be encouraged to write their approximating polynomials in ascending powers of $x$ so that they can more easily build them up term by term. For example, they can think of a degree $0$ polynomial as $p(x)=a$, a degree $1$ polynomial as $p(x)=a+bx$, a degree $2$ polynomial as $p(x)=a+bx+cx^2$ and so on. Once multiple students or pairs have one- or two-term polynomials for each function, they could start to compare their results and check if any approximations seem better than others. They can then continue to find further terms for their polynomials through numerical trial and improvement or they might consider what makes for a good approximation, such as similar gradient and curvature.
Key questions
What does each graph look like as you zoom in? Does this suggest a possible polynomial approximation?
Can you improve your approximation by changing the coefficients and/or adding more terms?