maximum of a polar equation

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xxMsJojoxx

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I don't understand how we are able to find the maximum of this equation, without using a graphing calculation.
r = 2 + 5cos θ

I know to find the zeros, we substitute r for 0.
But how do we find the maximums?
 
xxMsJojoxx said:
I don't understand how we are able to find the maximum of this equation, without using a graphing calculation.
r = 2 + 5cos θ
I know to find the zeros, we substitute r for 0.
But how do we find the maximums?
Click to expand...
First one must have some elementary knowledge of the cosine function such as:
1) the range is \(-1\le\cos(\theta)\le1\)
2) therefore, \(-5\le5\cos(\theta)\le 5\)
3) add two, \(-3\le2+5\cos(\theta)\le 7\).
From the above what is the maximum? Where (what \(\theta\)) causes the max?
 
xxMsJojoxx said:
I don't understand how we are able to find the maximum of this equation, without using a graphing calculation.
r = 2 + 5cos θ

I know to find the zeros, we substitute r for 0.
But how do we find the maximums?
Click to expand...
Please state the entire problem exactly as given to you.

An equation doesn't have a "maximum"; a function does. But since this is the polar coordinate equation of a curve, it is not clear whether you want the maximum value of r (greatest distance from the origin) as you seem to be saying, or something else (the highest point on the curve?).
 
Dr.Peterson said:
Please state the entire problem exactly as given to you.

An equation doesn't have a "maximum"; a function does. But since this is the polar coordinate equation of a curve, it is not clear whether you want the maximum value of r (greatest distance from the origin) as you seem to be saying, or something else (the highest point on the curve?).
Click to expand...

1603846363820.png

I think it's asking for the greatest distance form the origin. I'm just wondering how I would find maximum |r| on a test. I think it's just the maximum on a standard cosine graph, which is 1, when I graph y=cos θ on a graphing calculator and restrict the domain -pi to pi. --- Is that right?
 
Good -- it is indeed just looking for the greatest value of |r|, which in this case corresponds to the maximum of r. And, yes, that is when [MATH]\cos(\theta) = 1[/MATH], as they say. You can figure that out from familiarity with the graph of the cosine, or with the unit circle (or with a calculator, though it's better not to need that).
 
pka said just what I would have written. Since you still do not seem to understand what he said I will say it again. Hopefully this time you will look closer.

r = 2 + 5cos θ
we know that -1< cos θ < 1
So -5< 5cos θ < 5
Then -3< 2+5cos θ < 7
0< |2+5cos θ| < 7
so |r| is 7.
 
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