Those who have done either IGCSE Mathematics, IGCSE Further Mathematics or Additional Mathematics would have encountered this content. Otherwise it will be completely new!
3:: Identify increasing and decreasing functions.
5:: Find stationary points and determine their nature.
6:: Sketch a gradient function.
7:: Model real-life problems.
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As the second point gets closer and closer, the gradient becomes a better approximation of the true gradient:
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tangent
normal
Fro Exam Tip: A very common error is for students to accidentally forget whether the question is asking for the tangent or for the normal.
A function can also be increasing and decreasing in certain intervals.
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Leibniz’s
Newton’s
Lagrange’s
Original Function
First Derivative
Second Derivative
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convex
concave
(the same terms used in optics!)
i.e. the line curves in one direction before the point of inflection, then curves in the other direction after.
Technically we could label these either way round depending on where we view the curve from. What’s important is that the concavity changes.
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Gradient is 0.
Gradient is positive: initially increases but then decreases.
Gradient is 0, but positive before and after.
Gradient is positive: initially increases but then decreases.
Gradient is 0.
Gradient is negative. Initially decreases but then tends towards 0.
Gradient is positive, but tending towards 0.
We have a sheet of A4 paper, which we want to fold into a cuboid. What height should we choose for the cuboid in order to maximise the volume?
We need to introduce a second variable ourselves so that we can find expressions for the surface area and volume.
These are the two equations mentioned in the guidance: one for surface area and one volume.
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