MA1102 Calculus Tutorial 7

Topics Covered:

Increasing and decreasing functions. First and second derivative tests for relative extrema. Concavity. Point of inflection. Graph sketching.

Textbook: Chapter 7

Increasing function is synonymous with a function preserving the ordering '<'. Decreasing function is one reversing the ordering. These notions are global notions, that is, they apply to either a subinterval of a domain or the whole of a domain. Be careful about making conclusion of a global nature from a few local information. For example, f ' (a) > 0 does not guarantee that f is increasing around a, whereas f '(x) > 0 for all x in a connected domain (this is a global condition) implies that f is increasing in the connected domain. Concavity at a point is a local definition. For some text book, concavity only applies to a non-empty, nontrivial interval but not to a point. The local definition and the global definition are not the same. A consequence of the local nature of our definition is that if f '' (x₀) > 0, then the graph of f is concave upward at ((x₀, f (x₀)). But it does not necessary have to say anything about whether f ' (x) is increasing around x₀ . Beware. Stick to our definition. Most of the time we let the theorems do the work for us. For the serious minded, learn how these theorems come about and how they are proved using previously proven results such as Mean Value theorem, results about limits, etc. For graph sketching, which way the graph should bend can easily be determined by placing a tangent line there and follow the definition of concavity, either to place the graph above or below the tangent line. A point of inflection is just a point on the graph of the function, where the function is continuous and where there is a change in concavity.