Optimisation: some notes and commentary on how to approach these problems, at least as I see them.

Heinemann Higher Mathematics Ex 6Q Question 4:

Key points:

one-to-one correspondence

This topic will introduce some concepts which are useful for later parts of the course, so again please make a good effort to understand its content. While it could be thought of as a “helper” topic questions on this topic, in its own right, may also occur in the exam.

By the end of this topic you should understand

• what a set is
• what a function or mapping is
• what the domain of a function is
• what the range of a function is
• what a composite function is
• that f(g(x)) != g(f(x)) in general [!= means “not equal to”]
• one-to-one correspondence
• about the inverse function
• the effect of applying a function and then its inverse
• the graph of the inverse function is found by reflecting the graph of the function in the line y = x
• about exponential functions to a given base a
• about logarithmic functions to a given base a

The content of this topic is more abstract than the rest of the course but it will make topics to be encountered later on that are difficult, much easier to understand, and so this topic should not be dismissed as unimportant, or as a standalone topic.

By the end of this topic you should be able to answer the following questions

• Can you calculate the gradient of the line connecting two points eg (4,2) and (-5,7)?
• Positive lines slope in which direction?
• Negative lines slope in which direction?
• What is the gradient of a line parallel to the x-axis?
• What is the gradient of a line parallel to the y-axis?
• What can you say about the gradients of lines which are parallel to each other?
• How are the gradient of a line and the angle it makes in the positive direction of the x-axis related?
• How are the gradients of perpendicular lines related?

The knowledge and methods used in this topic occur in later topics so please make the effort to master them.