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Home » Mathematics » High School Pre-Olympiad (Ages 14+)
 
Number of Points of intersection.
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chetjan
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#1
Number of Points of intersection.
y=cos(pi/x) and y=100x+2
How many points of intersection do the graphs of y = cos(\frac {\pi}{x}) and y = 100x + 2 have?

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thejackchaner
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#2
Interesting
No idea
Looks interesting. Can you give us a hint of how big the number will be?

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ckck
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#3
Re: Number of Points of intersection.
y=cos(pi/x) and y=100x+2
chetjan wrote:
How many points of intersection do the graphs of y = cos(\frac {\pi}{x}) and y = 100x + 2 have?

Hint on what x can be
\cos(\frac{\pi}{x})=100x+2. cosine is between -1 and 1.
-1\le 100x+2\le 1
-\frac{3}{100}\le x\le -\frac{1}{100}
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chetjan
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Re: Interesting
No idea
thejackchaner wrote:
Looks interesting. Can you give us a hint of how big the number will be?

General Hint
The number of intersection will be less than 100
Specific Hint
The number of intersections will either be 67 or 68

PostPosted: Fri Nov 06, 2009 8:29 am  Back to top 
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chetjan
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#5
Answer
67
Solution
all you need to know is y(x) and y ' (x) for the given x.

When x=-0.03, the wavy line is at y=-0.5 and heading upwards steeply. The straight line is at y=-1.
The wavy line will then hit 1 when x=-1/34. It hasn't intersected yet.
By the time x reaches -1/36, when the wavy line is again at 1, it will have intersected the straight line twice.
By the time x reaches -1/38, it will have intersected it twice more.
And so on...
Between -1/98 and -1/100, it will intersect twice, and then intersect once more at the endpoint (-1/100, 1).

There are 33 even numbers from 36 to 100 inclusive. Double that to get 66. And add the last intersection to get 67.

To see that there are two intersections between -1/98 and -1/100, remember that, at (-1/100,1) the slope of the straight line is 100, and the slope of the wavy line is 0. So the straight line is under the wavy line for a brief moment beforehand.

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