Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the
parameter t varies.
In this 'mesh' of sine graphs, one of the graphs is the graph of
the sine function. Find the equations of the other graphs to
reproduce the pattern.
Sketch the graphs for this implicitly defined family of functions.
In chemistry, rates of reaction for complicated reactions are often approximated using simple polynomials. It is very useful to understand the qualitative nature of such algebraic representations. These concepts are explored in this task.
Consider the following algebraic forms for an approximate rate of a reaction $R$ in terms of $t$. When $R(t)$ is negative, it can be assumed that the reaction has either not started or has stopped.
$A: R_1(t) = -t +0.1t^3$
$B: R_2(t) = 2+ 2t - 2t^2$
$C: R_3(t) = 2 +t+0.1 t^2$
$D: R_4(t) = 5t - t^2$
$E: R_5(t) = t + t^2-0.1t^3$
$F: R_6(t) = -t + t^2$
Which of these start reacting immediately? Start slowly? Keep speeding up? Speed up to a peak and then slow down? Eventually stop? How would you best describe the reactions in words? Can you think of reactions which might be modelled by these sorts of equations?