For the following functions:
- Determine symmetry. Find roots and intercepts.
- Take the derivative and simplify.
- Find all the points where the derivative is zero or does not exist.
- Using the first derivative test, find the intervals where the
function is increasing and decreasing.
- Find the relative extremes.
- Find the second derivative and simplify.
- Find where the second derivative is zero or does not exist.
- Find the intervals where the function is concave up or down.
- List the inflection points;
- Plot points for the functions. Make sure you plot critical points, points of inflection,
intercepts, and endpoints of the domain (if it is an interval).
- Find the vertical and horizontal asymptotes (if they exist).
- Sketch the graph.
&a) f( x) = x^4 + 4x & &b) f( x) = \frac{x}{ x^2 + 4}\\
&c) f( x) = \frac{(1+x)\sqrt {1+x^2} }{ x} &
&d) f( x) = \frac{x^2 }{ x^2 -4} \\
&e) f( x) = \frac{x^2 }{ \sqrt {x^2 -2}} & &f) f( x) = x^3 + \frac{3}{ x}\\
&g) f( x) = 2\cos x \sin x -x, \text{on} [0,2\pi] &
&h) 2\tan x - \sec^2 x, \text{on} (0,\frac{\pi}{ 2})
\end{alignat*}
Tags: critical-points, max-min, curve-sketching, inflection-points, concavity, asymptotes, symmetry, higher-polynomial, rational, trig, root, composite, computation, sketching, a--a--g, t2, c2
I find that automaticity can be developed in the context of challenging problems. Today’s graphing calculators may render this problem pointless, but I think that it is empowering for students to really go through the steps to understand how to tackle graphing the old-fashioned way. The algebra involved in solving for the roots, critical points, etc. provides opportunities for reinforcing facility with factoring, etc.
– James Epperson 1998-08-09