Prospective Mathematics Teachers’ Knowledge of Asymptotes and Asymptotic Behaviour in Calculus

Abstract

This paper examines prospective mathematics teachers’ knowledge of asymptotes and asymptotic behaviour of functions in calculus. They are university students and future facilitators of knowledge in upper secondary education. We constructed a reference epistemological model to describe the knowledge about asymptotes for upper secondary and university education and explored prospective teachers’ relations to targeted knowledge. The study was conducted within the Anthropological Theory of the Didactic as a suitable framework to analyse and interpret knowledge of mathematical notions. Prospective teachers participated in three questionnaires with open, non-routine questions on different aspects of the notion of asymptotes. An analysis showed their knowledge was fragmented and their work relied heavily on algebraic manipulation and memorized formulas from calculus. The results indicated that knowledge of asymptotes and asymptotic behaviour is a potentially powerful context for developing knowledge related to the limits of functions in calculus.

Appendix

Question 1.1 Sketch a graph of the function $f(x)=\frac{2x-1}{x-1}$. Explain.

Question 1.3 It is expected that the percentage (expressed as a decimal) of viewers who will respond to a commercial message for a new product after t days, behaves according to the formula o(t) = 0.7 – 2^–t.

(a) Represent the given relationship o(t) graphically.

(b) What is the expected percentage of viewers who will respond to the commercial message after 7 days?

(c) Describe the behaviour of the expected percentage of viewers who will respond to the commercial message as the days pass.

Question 2.2 (a) Read the following assertion:

“The line y = kx + l is the slant asymptote of a function f if $\underset{x\boldsymbol{\to}\infty }{\lim}\left(f(x)-kx-l\right)=0$.”

Why is this true for the slant asymptote? How does that assertion fit with your description of an asymptote in the answer to the question 2.1?

(b) Explain the formulas $k=\underset{x\to \infty }{\lim}\frac{f(x)}{x}$ and $l=\underset{x\to \infty }{\lim}\left(f(x)- kx\right)=0$ for coefficients k and l of the function’s f slant asymptote y = kx + l.

(c) What are the conditions on the function f for the line x = a to be its vertical asymptote?

Question 2.3 The graph of the function f given by $f(x)=x+\frac{1}{x}$ is depicted in Figure 8a. Determine the equations of the asymptotes of the function f.

Fig. 8

Figures from the questionnaires

Question 2.4. The function, apart from a line, can have other curves as asymptotes. In Figure 8c, the function graph is depicted by a full line and the function’s asymptotic parabola by a dotted line.

(a) Express, using mathematical symbols, the requirement for a parabola given by y = ax2 + bx + c to be an asymptotic curve of the function f.

(b) How would you find the coefficients a, b and c of the asymptotic parabola of function f?

(c) Determine the equation of the asymptotic parabola of the function f given by $f\left(x\right)=\frac{x^3-3x^2+1}{x-1}$, according to its graph depicted in Figure 8b.

Question 3.1 In Figure 8c, a part of the graph of the function f given by $f(x)=3x-1+\frac{\mathit{\sin}\left(4\pi x\right)}{x}$ is depicted by a full line. Determine the equation of the line depicted by the dotted line in Figure 8c. Explain.