Curve sketching using derivatives from "summary" of Differential Calculus by S Balachandra Rao
To sketch the curve of a function using its derivatives, one must first find the critical points. These points are obtained by setting the derivative equal to zero and solving for x. The critical points can also be found by determining where the derivative does not exist. Once the critical points are identified, the next step is to determine the intervals where the function is increasing or decreasing. This can be done by analyzing the sign of the derivative in each interval. If the derivative is positive, the function is increasing; if it is negative, the function is decreasing.
After determining the increasing and decreasing intervals, one must locate the local maximum and minimum points. These points occur at critical points where the function changes from increasing to decreasing, or from decreasing to increasing. To find the concavity of the curve, one must analyze the sign of the second derivative. If the second derivative is positive, the curve is concave up; if it is negative, the curve is concave down.
Next, it is important to identify points of inflection where the concavity changes. These points can be found by setting the second derivative equal to zero and solving for x. Points of inflection occur at critical points where the concavity changes from positive to negative, or from negative to positive. By following these steps and analyzing the behavior of the function using its derivatives, one can sketch the curve accurately and understand its characteristics in detail.
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