Optimisation

Optimisation questions involve finding the best solution to a problem. Usually, that is the maximum or minimum value of a function. Questions on this topic require you to take information from a practical problem and write this as a function, then use differentiation to find the maximum or minimum value of this function by solving where the gradient is equal to zero. Often questions on this topic seem to have two variables, but you need to take a piece of information that fixes one of the variables and substitute this to create a function with one variable (although HL students might have to use Implicit Differentiation!)


Key Concepts

On this page, you should learn about

  • solving optimisation problems using differentiation

Essentials

The following videos will help you understand all the concepts from this page

Optimum Can

Optimum Can

In the following video we look at an example of using differentiation to find the best possible can for a drinks manufacturer.

A drinks manufacturer wants a design for a new can for their mini drinks collection, volume=170ml.Your job is to find the can that will contain a volume of 170ml = 170cm3 ≈54πcm3 and that will minimize the amount of aluminium used.

It might help you to visualize the can by playing with the following applet

Notes from the video

Optimisation - Rectangle in a Graph

The following is an example of an optimisation problem

A rectangle ABCD is drawn so that its lower vertices are on the x-axis and its upper vertices are on the curve \(f(x)=\sqrt{6-3x^2}\) as shown in the following diagram

Let OA = x
The area of this rectangle is denoted by A.

(a) Write down an expression for A in terms of x.

(b) Find the maximum value of A.


You may find the following applet useful to get an understanding of the problem.

Notes from the video

 

 

 

Test Yourself

Here is a quiz that practises setting up the problem before you use Calculus to solve. You are strongly recommended to have something to write with whilst doing these questions. You may want to draw sketches of shapes and graphs to help you think through the problems. You are not required to do any Calculus in this quiz!


START QUIZ!

Exam-style Questions

Question 1


Hint

Full Solution

Question 2

A container is made from a cylinder and a hemisphere. The radius of the cylinder is r m and the height is h. The volume of the container is \(45\pi\)

a) Find an expression for the height of the cylinder in terms of r

b) Show that the surface area of the container, \(A=\frac{5 \pi r^2}{3}+\frac{90 \pi}{r}\)

c) Hence, find the values of r and h that give the minimum surface area of the container

* Volume of a sphere = \(\frac{4}{3}\pi r^3\)

** Surface area of a sphere = \(4\pi r^2\)


Hint

Full Solution

 

Question 3

The diagram below shows the graph of the functions f(x) = sinx and g(x) = 2sinx

A rectangle ABCD is placed in between the two functions as shown so that B and C lie on g , BC is parallel to the x axis and the local minima of the function f lies on AD.

Let NA = x

a) Find an expression for the height of the rectangle AB

b) Show that the area of the rectangle, A can be given by A = 4xcosx - 2x

c) Find \(\frac{dA}{dx}\)

d) Find the maximum value of the area of the rectangle.

Hint

Full Solution

 

MY PROGRESS

How much of Optimisation have you understood?