Increasing/Strictly Increasing

[Daenerys Targaryen]

So we have approached differential calculus (in methods) and we have come across terminology 'increasing' and 'strictly increasing'. What is the difference between the two.

Put into context:

Where is this curve strictly increasing?

[Professor Polonsky]

bcub3d's spot on there - I originally got my stuff mixed up.

Basically, as long as the curve is not decreasing, it can be said to be strictly increasing (and vice versa).

[b^3]

http://www.vcaa.vic.edu.au/Documents/bulletin/2011AprilSup2.pdf

A function is said to be strictly increasing when implies for all and
in its domain. This definition does not require to be differentiable, or to have a non-zero
derivative, for all elements of the domain. If a function is strictly increasing, then it is a
one-to-one function and has an inverse function that is also strictly increasing.

That is you would have to say, include the turning points as well as the values of for which the gradient is negative. Say we had the function , then for all values of greater than , , as the function is increasing, the value will always be higher than the value at . But if we were asked where the function is increasing, then you'd find where the gradient is positive, i.e. not include .

So the difference, one may include the turning point (go back to the definition in the quote), and the other ('increasing') will not (as the gradient isn't positive).

EDIT: Also for past exams where it came up, check Methods 2009 Exam 2 ER Q1 http://www.vcaa.vic.edu.au/Documents/exams/mathematics/2009mmCAS2-w.pdf

EDIT2: Refer to Kamil's post.

[Daenerys Targaryen]

If that defines strictly then what is the difference of increasing?

[Professor Polonsky]

As bcub3d stated above, the function would be said to be strictly increasing at a turning point or a stationary point of inflexion. That is because there are no points with a lower x-value which have a higher y-value.

However, the function is not increasing at that point, as its gradient is zero.

Take .



The function is strictly increasing for its entire domain, as as long as .

However, it is not increasing at , as its gradient there is zero.

[kamil9876]

Not sure what they teach in VCE these days, but f(x)=x^3 is strictly increasing everywhere. The notion has NOTHING to do with differentiability, even the piecewise linear function for and for . This is strictly increasing, even at 0, even though it isn't differentiable there. Differentiability is merely a tool that can be used to determine when the function is increasing etc.

The definition should be: "f is strictly increasing if whenever we have " and "f is increasing if whenever we have "


For some reason this question is asked all time time on AtarNotes for the past 3 years.

[Professor Polonsky]

"f is increasing if whenever we have "

So is increasing at ?

[BubbleWrapMan]

kamil is correct, x^3 is increasing everywhere

[humph]

Not sure what they teach in VCE these days, but f(x)=x^3 is strictly increasing everywhere. The notion has NOTHING to do with differentiability, even the piecewise linear function for and for . This is strictly increasing, even at 0, even though it isn't differentiable there. Differentiability is merely a tool that can be used to determine when the function is increasing etc.

The definition should be: "f is strictly increasing if whenever we have " and "f is increasing if whenever we have "


For some reason this question is asked all time time on AtarNotes for the past 3 years.

This is still terrible terminology; I would say that "f is nondecreasing if whenever we have ". To me, increasing means strictly increasing.

Then again, I often write when I really mean , so who am I to complain?

[kamil9876]

^ I agree on all points (even the strict inclusion thingy) but I was trying to help the OP by sticking to their terminology. But I do think "terrible" is a bit of a stretch.

[BubbleWrapMan]

I'm pretty sure the term exists for the instances where we only really care that a function is non-decreasing, though that might have been a better term for it.