Introduction to Sketching Graph of Function
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Now, when we know methods of differential calculus let's consider question of sketching graph of the fucntion.
Firstly, consider continuous on finite interval `[a,b]` function `y=f(x)`. Here we are mainly interested in precise characteristic of change of function.
Before knowing methods of differential calculus we could only sketch graph of function by taking sufficiently large number of arbitrary points and connect the with lines. But this method is not applicable for practical purposes because it requires calculating of very large number of coordinates. Another disadvantage of this method is that since we take arbitrary points, we can miss some "important" points and graph will be sketched partly incorrectly.
Now suppose that function `y=f(x)` has finite derivative (except possibly at finite number of points). Then methods of differential calculus allow us to determine some number of "important" points, typical for this function. Using these points we already can draw graph with some precision.
First of all under "important" we mean points of extrema. However, we can add points where tangent line is horizontal or vertical, even if these points are not extrema.
When these points are drawn it is often enough to sketch the graph. This sketch represents behaviour of the function very well. It represents intervals where function is increasing and where decreasing, where rate of change of function is 0 (`f'(x)=0`) and where it is infinite (`f'(x)=+-oo`).
If we want to sketch even better graph, we need to determine intervals of convexity and concavity together with inflection points.