HSC Stuff > HSC Mathematics Extension 1
3U Maths Question Thread
Happy Physics Land:
--- Quote from: foodmood16 on February 09, 2016, 09:01:07 pm ---Hey I have a test coming up and I am having trouble with ext locus and parabola. The question is
Find the equation of the chord of contact AB of tangents drawn from an external point (x1, y1) to the parabola x2 = 12y
Thankyou :)
--- End quote ---
Hey foodmood16,
I did get an answer to your question, it was simply a matter of substituting into the equation of chord of contact. Just double check with your answer just in case I got it wrong (when questions are too easily solved I feel a bit intimidated). But yes great question doe because just about everyone tends to forget about this chord of contact equation.
(sorry for the dodgy diagram l drew)
Best Regards
Happy Physics Land
Neutron:
Hey! I was just practicing some curve sketching and I was wondering for curves that look like these:
y= 3x/(x^2+9)
How would you know the shape it forms? Cause really all I knew was that it intercepts at the origin, is odd and has a horizontal asymptote at 0.. Thanks, sorry if this sounds dumb it's just that it seems a bit dodgy how I'm assuming it looks like a sorta sideways s (sorry I can't really describe and I don't know how to draw/paste it in here).. Thank you!
Neutron
Happy Physics Land:
--- Quote from: Neutron on February 10, 2016, 06:23:32 pm ---Hey! I was just practicing some curve sketching and I was wondering for curves that look like these:
y= 3x/(x^2+9)
How would you know the shape it forms? Cause really all I knew was that it intercepts at the origin, is odd and has a horizontal asymptote at 0.. Thanks, sorry if this sounds dumb it's just that it seems a bit dodgy how I'm assuming it looks like a sorta sideways s (sorry I can't really describe and I don't know how to draw/paste it in here).. Thank you!
Neutron
--- End quote ---
Hey Neutron,
This is a really good question that can be typically encountered in both 3 unit and 4 unit exams. My solution here is more for the 3unit students that have not done as much graphing as the 4 unit students. If you are a 4 unit students, another alternative of approaching this question that I do recommend is through sketching x^2 + 9 first (which is just a parabola), then sketch 1/(x^2+9) (with x=0 as the asymptote and using special values such as when y = 0, 1 or -1 to help you, these are tricks that you would have learnt in sketching reciprocal functions), and in the end use multiplication of ordinates (again using special points where y = 0, 1 and -1) to sketch the graph.
My approach here is similar to yours. Except I have found the limits as x approaches infinity and I have constructed a table of values to help myself to see what the shape will resemble. In this table of values I only used positive values of x because since this is an odd function, you can simply turn the graph on positive side of x-axis by 180 degrees anticlockwise to find the graph on the negative side of the x-axis. I have also circled when x=3, y =1/2 as the maximum turning point in the positive graph through observing the table of values. Im assuming here that the question does not want you to use calculus, so we can only infer that there is a max tp. at (3,1/2). If you can use calculus, then the maximum and minimum turning points can be easily found and the curve would have been very easy to draw.
If you have any concerns or confusions please dont hesitate to ask! :D
Best Regards
Happy Physics Land
Neutron:
Thank you so much HPL! Yeah yeah it makes sense now.. One more thing though, how do you know whether a point is a point of discontinuity (open circle) or an asymptote?
Happy Physics Land:
--- Quote from: Neutron on February 10, 2016, 07:16:18 pm ---Thank you so much HPL! Yeah yeah it makes sense now.. One more thing though, how do you know whether a point is a point of discontinuity (open circle) or an asymptote?
--- End quote ---
Another very good question indeed, and this definitely is something that will be hard to distinguish between.
When you are provided with an equation, you should become suspicious of a point of discontinuity under two situations:
1. when a y-value of your curve approaches zero as x approaches the asymptote but the curve looks like as if it's going to intersect the asymptote
2. when you have f(x)g(x) but the domains are not the same (a typical example is xlnx, where domain of y = x is defined for all real x, but the domain for y = lnx is only defined for x>0)
A point of discontinuity is essentially when the function is undefined for both x and y-values of this particular point. For example, lets raise the y = xlnx again. x= 0 would be undefined for lnx because x>0, y = 0 is undefined because it is impossible to get a result of 0 from a log function. Hence the point (0,0) would be a point of discontinuity. When you become suspicious that there is a point of discontinuity, substitute in x and y-values of that point to confirm whether or not the point is discontinuous.
Finding a discontinuous point is not easy and it will take practise to develop an instinct to become suspicious of the existence of such points. In general, whenever you see lnx involved, you should keep an eye out on possible points of discontinuity.
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