By the time curve sketching is formally introduced, students have already seen a few “purposeful” examples; they are listed below as links.
–Putnam Problem, Four Towns, e^(Pi) vs (Pi)^e, f(x)=x^x (L’Hospital’s Rule)
Through these examples (and others), first and second derivatives have been points of discussion with respect to identifying the location and nature of critical points. In addition to this, L’Hospital’s Rule will have been established along the way and thrown into the mix.
My formal introduction to curve sketching is introduced in the form of a problem solving scenario. An example of this is shown below.
Sum of a Number and its Reciprocal
Limits at Infinity & Infinite Limits
First and Second Derivatives
Graph of S(x) = x + 1/x
Click on the link provided to interactively determine the local extrema.
Graph of y=sin^2(2x)/x^2 (see L’Hospital’s Rule for indeterminate limits of this function)
Click on the link provided here to explore the relationships between y, y’ and y”.
Thanks for reading.
samuelson mathxp 