International Baccalaureate Mathematics

Syllabus Content

The second derivative. Graphical behaviour of functions, including the relationship between the graphs of f,f' and f"

Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.

Differentiation Video A reminder of how to differentiate different types of functions and how to find the equations of tangents and normals.

Differentiation Practise the technique of differentiating polynomials with this self marking exercise.

Here are some exam-style questions on this statement:

"Consider the cubic function $f(x)=\frac{1}{6}x^3-2x^2+6x-2$" ... more

"The following diagram shows part of the graph of $y=f (x)$" ... more

"The following diagram shows the graph of $f'$, the first derivative of a function $f$." ... more

"Let $f(x) = \frac{ln3x}{kx} $ where $ x \gt 0$ and $ k \in \mathbf Q^+ $." ... more

See all these questions

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

Calculus It is said that the word calculus comes from the Latin word for the small pebble used for counting and calculations. The two major branches, differentiation and integration, are studied by pupils only towards the end of their school days but does then form a major part of their studies. A course in calculus is a prerequisite for other, more advanced courses in mathematical analysis.

Furthermore

Official Guidance, clarification and syllabus links:

Use of both forms of notation, $\frac{d^2y}{dx^2}$ and $f''(x)$.

Technology can be used to explore graphs and calculate the derivatives of functions.

Link to: function graphing skills (SL2.3).

The second derivative, denoted as $ f''(x) $, provides insight into the concavity of the original function $ f(x) $. If $ f''(x) > 0 $, the function $ f(x) $ is concave upwards in that interval, indicating a local minimum if $ f'(x) $ changes sign. Conversely, if $ f''(x) < 0 $, $ f(x) $ is concave downwards, hinting at a local maximum if $ f'(x) $ undergoes a sign change. The points where $ f''(x) = 0 $ or is undefined are potential inflection points, where the concavity of $ f(x) $ might change.

Let's consider an example to illustrate this relationship:

Suppose $ f(x) = x^3 - 3x^2 + 2 $.

The first derivative, $ f'(x) $, represents the slope of $ f(x) $ and is given by:

$$ f'(x) = 3x^2 - 6x $$

The second derivative, $ f''(x) $, indicates the concavity of $ f(x) $ and is:

$$ f''(x) = 6x - 6 $$

From $ f''(x) $, we can determine that the function is concave upwards when $ f''(x) > 0 $ (i.e., $ x > 1 $) and concave downwards when $ f''(x) < 0 $ (i.e., $ x < 1 $). The point $ x = 1 $ is an inflection point.

This video on Optimization and Calculus Curves is from Revision Village and is aimed at students taking the IB Maths AA SL/HL course.

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