Question

what is curve sketching? explain the different steps involved in it?

Answer 1

uppose we are given continuous on function that is twice differentiable, except points where derivative doesn't exist or has infinite value.
To sketch the graph of the function, we need to perform the following:
Determine, whether function is obtained by transforming a simpler function, and perform necessary steps for this simpler function.
Determine, whether function is even, odd or periodic. This allows to draw graph of the function on some subinterval and then just reflect the result.
Find y-intercept (point ).
Find x-intercepts (points where ).
Find what asymptotes does function have, if any.
Calculate first derivative and find points where or doesn't exist, in other words find stationary points.
Use test (First Derivative or Second Derivative) to classify stationary points.
Find intervals where function is increasing () and where it is decreasing ().
Find points where or doesn't exist and test whether these points are points of inflection.
Find where function is concave up () and where it is concave down ().
After you've found "important" points calculate corresponding values of function at these points.
Add "control" points (some arbitrary points), if needed.
Draw "important" and "control" points and connect them by lines taking into account found behaviour of the function.
If function is even, odd or periodic then perform corresponding reflection.
If function is obtained by transforming simpler function, perform corresponding shift, compressing/stretching.
It is often convenient to draw all points, you've found, in the table.

Answer 2

curve sketching or curve tracing includes techniques that can be used to produce a rough idea of overall shape of a PLANE CURVE given its eqation without computing the large numbers of points required for a detailed plot.