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#1 2021-05-02 07:02:38
- mathland
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- Registered: 2021-03-25
- Posts: 444
Minimum & Maximum
Waves in deep water tend to have the symmetric
form of the function f (x) = sin x. As they approach shore,
however, the sea floor creates drag which changes the shape of the
wave. The trough of the wave widens and the height of the wave
increases, so the top of the wave is no longer symmetric with the
trough. This type of wave can be represented by a function such as
w(x) = 4/(2 + cos x)
(a) Graph w = w(x) for 0 ≤ x ≤ 4π.
(b) What is the maximum and the minimum value of w?
1. For (a), I gotta graph w(x).
2. To find the minimum and maximum value of w, I must do what exactly?
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#2 2021-05-02 07:07:12
Re: Minimum & Maximum
To find the minimum and maximum value of w, there are a few different approaches you could take.
The question is asking: what are the smallest and largest possible values of ?Here's a starting point: what are the smallest and largest possible values of ?Offline
#3 2021-05-03 09:12:53
- mathland
- Member
- Registered: 2021-03-25
- Posts: 444
Re: Minimum & Maximum
To find the minimum and maximum value of w, there are a few different approaches you could take.
The question is asking: what are the smallest and largest possible values of ?Here's a starting point: what are the smallest and largest possible values of ?
The largest value of cos (x) is 1.
The smallest value of cos (x) is -1.
Where are we going with this?
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#5 2021-05-03 11:23:53
- mathland
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Re: Minimum & Maximum
What are the smallest and largest values of 2 + cos(x)?
Let cos x = -1.
2 + cos x = 2 + (-1) = 1
Let cos x = 1
2 + cos x = 2 + 1 = 3
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#7 2021-05-03 11:47:58
- mathland
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- Registered: 2021-03-25
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Re: Minimum & Maximum
Yes -- finally, what are the maximum and minimum values of w(x)?
Maximum Value
w(x) = 4/(1)
w(x) = 4
Minimum Value
w(x) = 4/(3)
w(x) = 4/3
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#8 2021-05-05 08:25:48
Re: Minimum & Maximum
Correct.
One other approach you could have used is to calculate w'(x) to determine its stationary points, then evaluating w''(x) at these points to determine which of these is a maximum and which of these is a minimum. However, the approach taken above is a lot quicker and perhaps more intuitive.
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#9 2021-05-05 10:33:21
- mathland
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- Registered: 2021-03-25
- Posts: 444
Re: Minimum & Maximum
Correct.
One other approach you could have used is to calculate w'(x) to determine its stationary points, then evaluating w''(x) at these points to determine which of these is a maximum and which of these is a minimum. However, the approach taken above is a lot quicker and perhaps more intuitive.
Can you solve this problem using the other approach you talked about? I would like to see it done in terms of extra math study notes.
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#10 2021-05-05 10:56:55
Re: Minimum & Maximum
You'd need to:
-Differentiate w(x) with respect to x, to obtain an expression for w'(x)
-Find the values of x for which w'(x) = 0
-Substitute these values of x into your equation for w''(x) to determine which values of x correspond to a minimum and which correspond to a maximum
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#11 2021-05-05 12:20:25
- mathland
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- Registered: 2021-03-25
- Posts: 444
Re: Minimum & Maximum
You'd need to:
-Differentiate w(x) with respect to x, to obtain an expression for w'(x)
-Find the values of x for which w'(x) = 0
-Substitute these values of x into your equation for w''(x) to determine which values of x correspond to a minimum and which correspond to a maximum
Ok. Now, by w"(x) you mean the second derivative. Yes?
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#12 2021-05-05 16:18:08
- Mathegocart
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- Registered: 2012-04-29
- Posts: 2,226
Re: Minimum & Maximum
zetafunc wrote:You'd need to:
-Differentiate w(x) with respect to x, to obtain an expression for w'(x)
-Find the values of x for which w'(x) = 0
-Substitute these values of x into your equation for w''(x) to determine which values of x correspond to a minimum and which correspond to a maximumOk. Now, by w"(x) you mean the second derivative. Yes?
That's what it means, yes.
The integral of hope is reality.
May bobbym have a wonderful time in the pearly gates of heaven.
He will be sorely missed.
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#13 2021-05-05 19:36:20
- mathland
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- Registered: 2021-03-25
- Posts: 444
Re: Minimum & Maximum
mathland wrote:zetafunc wrote:You'd need to:
-Differentiate w(x) with respect to x, to obtain an expression for w'(x)
-Find the values of x for which w'(x) = 0
-Substitute these values of x into your equation for w''(x) to determine which values of x correspond to a minimum and which correspond to a maximumOk. Now, by w"(x) you mean the second derivative. Yes?
That's what it means, yes.
Ok. I can take it from here.
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