\( \DeclareMathOperator{cosec}{cosec} \)

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Here are some exam-style questions on this statement:

- "
*A function is given as \(f(x)=3x^2-6x+4+\frac3x,-2\le x \le 4, x\ne 0\).*" ... more - "
*Consider the graph of the function \(f(x)=7-3x^2-x^3\)*" ... more - "
*A child's play tent is made in the shape of half a cylinder. It is constructed from a fibreglass frame with material pulled tightly around it. The fibreglass frame consists of a rectangular base, two semi-circular ends and two further support rods, as shown in the following diagram.*" ... more - "
*A package is in the shape of a cuboid and has a length \(l\) cm, width \(w\) cm and height of 12 cm.*" ... more - "
*A particle moves in a straight line. During the first nine seconds the velocity, \(v\) ms*^{-1}of the particle at time \(t\) seconds is given by:$$ v(t) = t \cos(t+5) $$

*The following diagram shows the graph of v:*" ... more - "
*Consider the function \(f(x)=6 - ax+\frac 3{x^2},x\neq 0\)*" ... more - "
*The function \(f\) is defined for all \(x \in \mathbb{R}\). The line with equation \(y=5x+3\) is the tangent to the graph of \(f\) at \(x = 2\)*" ... more - "
*Consider the cubic function \(f(x)=\frac{1}{6}x^3-2x^2+6x-2\)*" ... more - "
*Consider the function \(f(x)=x^3-9x+2\).*" ... more - "
*Let \(f(x)=\frac{g(x)}{h(x)}\), where \(g(3)=36\), \(h(3)=12\), \(g'(3)=10\) and \(h'(3)=4\). Find the equation of the normal to the graph of \(f\) at \(x=3\).*" ... more - "
*The displacement, in millimetres, of a particle from an origin, O, at time t seconds, is given by \(s(t) = t^3 cos t + 5t sin t\) where \( 0 \le t \le 5 \) .*" ... more - "
*A circle with equation \(x^2+y^2=25 \) has centre \((0,0)\) and radius 5.*" ... more - "
*The function \(f\) is defined as follows:*" ... more - "
*Consider the functions*" ... more - "
*The following diagram shows part of the graph of \(y=f (x)\)*" ... more - "
*Moresum Soup is sold in cans with a capacities of 400ml each. Each can is in the shape of a right circular cylinder with radius \(r\) cm and height \(h\) cm.*" ... more - "
*Make a sketch of a graph showing the velocity (in \(ms^{-1}\)) against time of a particle travelling for six seconds according to the equation:*" ... more - "
*The function \(f\) is such that \(f(x) = \frac{\ln2x}{x^3} \) where \(x \gt 0\).*" ... more - "
*Consider the function \(f\) defined by \(f(x) = 25e^{x-5}\) for \(x \in \mathbb{R}^+\).*" ... more - "
*The following diagram shows the graph of \(f'\), the first derivative of a function \(f\).*" ... more - "
*The following equation defines a curve which passes through \( A( 2 \pi ,3 \pi)\)*" ... more - "
*Let \(f(x) = \frac{ln3x}{kx} \) where \( x \gt 0\) and \( k \in \mathbf Q^+ \).*" ... more - "
*Consider the function \(f(x)=\frac{20}{x^2}+kx\) where \(k\) is a constant and \(x\neq0\).*" ... more - "
*The north corridor at Addwell High School has a width of \(2 \, \text{m}\). There is a ninety-degree corner at point \(C\). Points \(A\) and \(B\) are variable points on the base of the walls such that \(A\), \(C\), and \(B\) lie on a straight line.*" ... more

Here are some Advanced Starters on this statement:

**Fence Optimisation**

Find the length of a rectangle enclosing the largest possible area. more**Maximum Product**

Two numbers add up to 10. What's the largest possible product they could have? more**Road Connections**

Design roads to connect four houses that are on the corners of a square, side of length one mile, to minimise the total length of the roads. more

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

This Optimisation video is from Revision Village and is aimed at students taking the IB Maths Standard level course.

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

If you use a TI-Nspire GDC there are instructions useful for this topic.

How do you teach this topic? Do you have any tips or suggestions for other teachers? It is always useful to receive feedback and helps make these free resources even more useful for Maths teachers anywhere in the world. Click here to enter your comments.