Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2021-05-02 07:02:38

mathland
Member
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?

Offline

#2 2021-05-02 07:07:12

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,432
Website

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

zetafunc wrote:

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?

Offline

#4 2021-05-03 10:57:05

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,432
Website

Re: Minimum & Maximum

What are the smallest and largest values of 2 + cos(x)?

Offline

#5 2021-05-03 11:23:53

mathland
Member
Registered: 2021-03-25
Posts: 444

Re: Minimum & Maximum

zetafunc wrote:

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

Offline

#6 2021-05-03 11:26:15

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,432
Website

Re: Minimum & Maximum

Yes -- finally, what are the maximum and minimum values of w(x)?

Offline

#7 2021-05-03 11:47:58

mathland
Member
Registered: 2021-03-25
Posts: 444

Re: Minimum & Maximum

zetafunc wrote:

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

Offline

#8 2021-05-05 08:25:48

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,432
Website

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.

Offline

#9 2021-05-05 10:33:21

mathland
Member
Registered: 2021-03-25
Posts: 444

Re: Minimum & Maximum

zetafunc wrote:

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.

Offline

#10 2021-05-05 10:56:55

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,432
Website

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

Offline

#11 2021-05-05 12:20:25

mathland
Member
Registered: 2021-03-25
Posts: 444

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 maximum

Ok. Now, by w"(x) you mean the second derivative. Yes?

Offline

#12 2021-05-05 16:18:08

Mathegocart
Member
Registered: 2012-04-29
Posts: 2,226

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 maximum

Ok. 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.

Offline

#13 2021-05-05 19:36:20

mathland
Member
Registered: 2021-03-25
Posts: 444

Re: Minimum & Maximum

Mathegocart wrote:
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 maximum

Ok. Now, by w"(x) you mean the second derivative. Yes?

That's what it means, yes.

Ok. I can take it from here.

Offline

Board footer

Powered by FluxBB