Refer to Exercise 85 . (a) Determine the directions in which I has maximum or minimum values. (b

Refer to Exercise 85 . (a) Determine the directions in which...

Key Concepts

Trigonometric Functions

Trigonometric functions are periodic functions that model oscillatory behavior, commonly found in many physical phenomena like waves and rotations. Their inherent periodicity makes them suitable for problems defined on intervals such as [0, 2?], where identifying repeating maximum and minimum values is essential.

Optimization in Functions

Optimization in functions is the process of finding the input values that yield maximum or minimum outputs. It typically involves techniques such as differentiation to locate critical points, which represent potential optimal values in the context of the function’s behavior.

Graphical Analysis

Graphical analysis involves plotting the behavior of a function over a specified interval to visually identify key features such as peaks, troughs, and points where the function attains a given value. This method is particularly useful for approximating values when analytical solutions are complex or unavailable.

Critical Points

Critical points refer to values in the domain of a function where its derivative is zero or undefined, leading to potential local maxima, minima, or points of inflection. This concept is crucial for determining the specific directions in which a function, such as an intensity distribution, achieves its extreme values.

Transcript

00:01 I'm going to do the two problems that relate to the radio signal intensity.

00:07 Two it together because they both basically they refer to one another.

00:14 So what we have, and maybe i'd actually just draw a little bit of sketch here.

00:19 So we got the radio tower here and another one here.

00:23 And then as we go in a circle around this, a circle here.

00:31 We get different intensities and basically theta is measured this way.

00:40 So they ask us to look at a few different values of theta in their first problem.

00:47 So they give us what theta is in terms of some reference intensity.

00:54 That turns out to be what the intensity is when theta equals zero.

00:58 So right out here.

00:59 And we see that when theta equals 0, this is 0.

01:04 So this is 0.

01:06 So this is 1.

01:07 So this is 2.

01:08 So this is 1.

01:09 So the intensity at theta equals 0 is just i .0, the reference intensity.

01:15 So that's where it actually winds up being a peak here.

01:19 And also when, obviously, when you get to theta equals pi, it's also a peak over here.

01:27 So basically we have symmetry.

01:28 About this.

01:33 Let's see here.

01:34 Then when they ask is for pi over three, which is like somewhere around here.

01:42 And we plug that in, and we get, this thing is close to minus one, so this thing is close to zero, and then we divide by two and we get 0 .04 -36 times the intensity.

01:58 And so, you know, that we're right, pie over, you know, we're in, this area here we're in this very flat zone here and likewise we would have if we plugged in minus pi over three we would have the same value and now if we go to pi over seven which means let's see you know we're probably this is pi over four pi over eight and also pi over seven is maybe right in here somewhere and so pi over seven is going to be like say right around here and we see that we wind up with plugging that in we get 0 .603 times the peak intensity.

02:40 So basically we're at 60 % of the intensity as we get.

02:46 So you know if we could actually plot this on this kind of axis and we'd see that it was zero around here and then it comes up.

02:55 Well let's see here that's it's zero kind of and then it comes up.

03:01 And then it gets up heat and then it comes down to back 0 and then it's, you know, that's what it would look like kind of on this, you know, if we wrapped it around this circle.

03:13 So that's what the values are at those three points they asked in the first problem.

03:17 And then the second problem, they discerned, they determine the directions in which i has a maximum minimum value.

03:22 Well, you can see here obviously that when theta equals 0 or pi, or minus pi, which is the same point obviously, we have, you know, it's i is i not, the reference, whatever the reference value is.

03:43 And so it's a maximum.

03:45 So if we plot i over i not, this point here is one.

03:49 And let's see, we have, we can see that, you know, when cosine is the cost, when this cosine function is minus one, this goes to z.

04:00 And so that occurs when theta equals minus pi over two or pi over two.

04:06 So it's actually, you know, actually technically zero at these two points.

04:11 Over this range it's not exactly zero, but obviously it's very close to zero in these ranges.

04:16 So, you know, in the range, you know, from here to here, say, you know, you're getting very little signal propagating out this way and out this way...

Please give Ace some feedback