Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Identify turning points using the first derivative.
• Classify turning points as maxima, minima, or saddle points.
• Use the Second Derivative Test.
• Sketch the graph of a function.

Lesson summary

In our study of functions, we often want to find the "best" or "most extreme" values. For example, a business owner in Makola Market wants to know the production level that will give the maximum possible profit. An engineer designing a bridge wants to calculate the maximum load it can bear. A farmer wants to know the minimum amount of fencing needed to enclose a certain area. Calculus, specifically the use of derivatives, provides a powerful tool to solve these optimization problems.

Lesson notes

A. What are Stationary Points?

A stationary point on a curve is a point where the gradient (or slope) of the curve is zero. At this point, the function is momentarily "flat" — it is neither increasing nor decreasing.

To find stationary points, we use the fact that the first derivative, `dy/dx` or `f'(x)`, represents the gradient of the function. Therefore, at a stationary point: `dy/dx = f'(x) = 0`

There are three main types of stationary points we will study: Local Maximum: A point that is higher than all the nearby points on the curve. The curve looks like an upside-down bowl (∩) at this point. Local Minimum: A point that is lower than all the nearby points on the curve. The curve looks like a regular bowl (∪) at this point. Saddle Point (or Point of Inflexion): A point where the gradient is zero, but it is neither a maximum nor a minimum. The curve flattens out and then continues in the same general direction. It looks like an 'S' shape lying on its side. B. The Second Derivative Test: The Tool for Classification

Evaluation guide

• Use the second derivative of a function to classify the maximum, minimum and saddle
• point of that function and perform curve sketching.
• Talk for Learning, Think -pair- share, Experiential Learning and Group
• Work/Collaborative Learning.