Math Is Fun Forum

Getting a 5 on the first die and a 5 on the second die is the same as getting a 5 on the second die and a 5 on the first die.

What is the limit of your function as x approaches 2?

Hi Sara,

Thanks for stopping by -- have you considered registering an account with us?

Bobbym sadly left us last year. We all miss him very much. There is a thread dedicated to him here: http://www.mathisfunforum.com/viewtopic.php?id=23912

The key word here is that we want to find the largest time interval where the percentage growth was an increasing function of time. Imagine plotting a graph with years on the x-axis and percentage growth on the y-axis -- what would it look like?

No, it's -1/4: the gradient of a line and the gradient of the line perpendicular to it must multiply to make -1.

Correct! And you can check your answer by substituting x = 1 into your answer to get y = 5.

Now for the perpendicular line passing through (1,5). The line y = 4x + 7 has a gradient of 4. What is the gradient of the perpendicular line?

No, you need to find two lines: one which is parallel to y = 4x + 7, and one which is perpendicular to it, each of which pass through (1,5).

First, let's examine what is meant by the word 'increasing'. How can you look at a graph and immediately tell if it is increasing or not? One way is to look at the slope (or, if you've done any calculus, the 'derivative', which tells you the slope of the function at every point). If the slope at that point is positive, then the function is increasing. If it is negative, it is decreasing. If it is 0, then it's stationary. Now let's try to understand what is meant by 'monotone'.

This is an example of a function which is monotonically increasing. Notice that the graph is always 'heading upwards' (think about the tangents to this curve for a more accurate description). This happens until we get to the middle, where the graph resembles a horizontal line. It's not increasing along this part -- but it also isn't decreasing. In other words, if something is monotonically increasing, then it doesn't decrease.

This is an example of a function which is monotonically decreasing. Notice that the graph is always 'heading downwards', except when we get to the horizontal part of the curve. In other words, if something is monotonically decreasing, then it doesn't increase.

That's a function which isn't monotone at all -- do you see why?

As for 'strictly increasing', then this means that the function must always be increasing -- it won't have any 'horizontal parts'. Likewise for 'strictly decreasing'.

There are formal mathematical definitions for all of these terms, by the way -- but I want to make sure that you understand what the picture looks like first. Does this help?