I spotted this little problem on Twitter (see Figure 1):
![]() |
Figure 1 |
The proximity of to should alert us to the possibility that either or is a solution. After that, the corresponding and values fall into place. However, let's suppose that we didn't spot this and I have to confess I didn't. In that case, we follow the conventional method.
At first sight, it looks like you will end up with a quintic equation after substituting into and there is no general formula for solving quintic equations. However, the terms cancels out nicely leaving us with a quartic equation. Let's see how: Thus and or and . Using GeoGebra, this can be confirmed visually as shown in Figure 2:
Differentiation reveals a little more about the graph of : It can be seen that is undefined when and thus the graph is vertical at this point. If we zoom in a little, this becomes clearer (see Figure 3). There is a point of horizontal inflection when .
![]() |
Figure 3 |
This was a useful little exercise as I got to revisit the binomial expansion for
In this graph, the vertical tangent occurs at and this got me thinking about the area under the curve between 0 and 1. In other words, what is the value of: Using SageMathCell, the integral evaluates to: Now I vaguely remember coming across a so-called function that is in some way, I think, related to the function. Further investigation revealed that they are indeed related. In fact: This result looks about right but what is going on with this function? Well, the beta function is actually an integral between 0 and 1 involving two different values, let's say and , so that we have: If we can get our integral into that form then we can integrate it. Firstly, we make a substitution. Let and so and . We can write and the integral now transforms into: So this was quite an interesting post that led me in some unexpected directions. I also was interested in the area between the curves and in the range from 0 to 1. See Figure 5 where eq2 is and the curve is a tangent to the circle at and .
Clearly, the area between the two curves from is given by . What I've learned from all this is that the function is a very useful tool to have in one's integration armoury.
No comments:
Post a Comment