Lesson 63 — Curve Sketching

Determine the intervals of increase/decrease and concavity of $$.

For $$, determine: intervals of increase/decrease, local extrema, concavity, and inflection points.

Completely analyze $$: domain, asymptotes, extrema, concavity.

Sketch $$ (Gaussian curve): domain, extrema, inflections, asymptote.

Analyze $$ for $$: domain, extrema, concavity.

Sketch $$: domain, vertical and horizontal asymptotes, extrema.

Determine the inflection points and intervals of concavity for $$.

Completely analyze $$: extrema, inflection, asymptote.

Analyze $$: domain, minimum, concavity.

Determine the slant asymptote and sketch $$.

Determine intervals of increase, local extrema, and concavity of $$ on $$.

The correct definition of an inflection point is:

"Concave up on an interval" means that:

Completely analyze $$.

Analyze $$: extrema, concavity, behavior at infinity.

For $$ on $$, determine local extrema, inflections, and sketch.

Sketch $$: domain, asymptotes, increase, concavity.

Completely analyze $$: extrema, inflections, concavity, symmetry.

Determine domain, asymptotes, and extrema of $$... (correct to $$ — even function with asymptotes at $$).

Completely sketch $$.

Analyze $$ for $$: extrema, concavity, intercepts.

Determine the slant asymptote and extrema of $$.

Prove that if $$ on $$, then the graph of $$ lies above any tangent line on $$ (a concave up function lies above its tangents).

Show that every cubic function $$ with $$ has exactly one inflection point, and that the graph is symmetric about this point.

Completely sketch $$ for $$, including the limit as $$.

Completely analyze $$: extrema, inflections, concavity, symmetry.

Analyze $$: domain, maximum, inflection, asymptote.

Analyze the extrema and symmetry of $$.

Sketch $$: asymptotes, extrema, parity.

Analyze the family of curves $$ with $$: determine extrema and inflections as functions of the parameters. How do $$ and $$ control the curve's shape?