Calculus - Behaviour of the function and curve sketching — Study curves

About this lesson

This unit focuses on the behavior of functions and curve sketching in calculus. It begins by introducing the concepts of convexity (convex upward and convex downward) and how to determine the intervals of convexity using the second derivative. The text explains that if the second derivative is positive on an interval, the curve is convex downward, and if negative, it is convex upward. It also covers inflection points, where the direction of convexity changes, and provides conditions for their existence, including the requirement of a tangent at the point. The unit then presents the second derivative test for finding local maximum and minimum values, stating that if the second derivative is negative at a critical point, it is a local maximum, and if positive, a local minimum. Finally, the unit demonstrates a systematic approach to sketching curves of polynomial functions, involving steps such as checking symmetry, finding critical points, determining intervals of increase/decrease and convexity, locating inflection points, and plotting intercepts. Examples and exercises are provided throughout, including piecewise functions and applications of the second derivative test.