Forest Defoliation To estimate the amount of defoliation caused by the gypsy moth during a year,

Forest Defoliation To estimate the amount of defoliation...

Key Concepts

Calculus and Optimization

Calculus, particularly the use of derivatives, is employed to determine rates of change and to identify extrema in functions. In this scenario, taking the derivative of the defoliation function and finding its maximum helps identify the point at which the percentage defoliation is increasing most rapidly, illustrating the concept of optimization in mathematical modeling.

Inverting Functions

Inverting a function means solving for the independent variable when given an output value. This process is essential for problems where one needs to determine the original input, such as estimating the number of egg masses based on an observed defoliation percentage.

Function Evaluation

Function evaluation entails substituting a particular value into a function to determine the corresponding output. This is important for making predictions or interpreting the model in practical scenarios, such as estimating the percent defoliation from a known count of egg masses.

Graphing and Visualization of Functions

Graphing functions helps to visualize how the dependent variable behaves across different values of the independent variable. This is crucial for understanding trends, inflection points, and overall behavior of models such as the defoliation function, which may have asymptotic behavior or regions of rapid change.

Exponential Functions

Exponential functions involve a constant base raised to a variable exponent and are used to model growth or decay processes. In this context, the exponential term in the defoliation function controls how rapidly the effect of egg masses on defoliation changes, capturing non-linear relationships common in biological and environmental processes.

Transcript

00:01 Alright, this problem starts out with telling you to graph this function.

00:05 So go into y sub 1 and make sure that when you type this in, i did it in alpha y equals, that you have 300 over 3 plus 17 e to the negative 0 .0625 x.

00:23 Now, when you go to graph this, it's not going to show on your viewing window.

00:28 So you're probably going to want to change your viewing window even before you hit go.

00:34 Make your x min 0, make your x max 100.

00:41 I counted by tens, just to keep it clean a little bit.

00:47 Y min 0, y max 100, and i kept that scale also at 10.

00:56 And you should get a pretty nice picture of this on your viewing screen.

01:01 It's a nice logistic growth model.

01:06 It kind of looks like that.

01:08 So you get the idea of what it's supposed to look like for part a.

01:14 Part b wants to know, let's get the wording here, estimate the percent of defoliation if 2 ,000 egg masses are counted.

01:26 Alright, so 2 ,000 egg masses, that just means x is 2.

01:34 So on your graph, you're going to want to hit second and then the button for trace.

01:43 And it brings up your calculate menu.

01:46 And you want to choose option number one value.

01:49 And then go ahead and type in a 2.

01:53 And it'll come up x equals 2 on your screen.

01:57 And voila, all of a sudden you'll get an answer of 16 .6644, which you'll have to convert to 16 .7%.

02:10 Then part c says, estimate the number of egg masses that exist if you observe that approximately two -thirds of a forest is defoliated.

02:21 So that means you're going to let a y value equal 66 .67%, right? because that's the kind of answer we're getting.

02:28 So you want to go into y sub 2 now and let that equal 66 .67.

02:36 When you do that, go ahead and graph it.

02:38 You'll get a nice solid straight line coming across in there somewhere.

02:42 And we're going to look for the intersection point.

02:45 To do that, on your calculator again, you want to hit that second trace again.

02:52 But this time, choose option number five for intersect.

03:00 Now it gives you choices.

03:01 It says first curve, second curve, guess, and all that stuff.

03:04 Because these are the only two things on there.

03:06 All you have to do is press enter three times.

03:14 Press enter three times.

03:15 That's a really ugly three i already said for you.

03:17 Press enter three times.

03:19 All right? and then you'll get an answer for an x value of 38 .846372.

03:33 And again, remember, those x values are in thousands.

03:38 So basically i want to multiply by a thousand, which is the same as moving my decimal point three places to the right.

03:46 So that's a grand total of 38 ,846 eggs, egg masses.

03:52 I think they are.

03:57 All right.

03:58 Now part d requires the most work.

04:01 It says use calculus to estimate the value of x for which y is increasing most rapidly.

04:07 So what that's going to mean is in here you're talking about how fast your slope is changing.

04:12 So changing, changing, changing.

04:15 Well, in here it's going to change the most.

04:18 And that's where that critical point is.

04:19 That's basically where your poi is.

04:22 All right? so we have to figure out what that poi is.

04:26 Well, in order to find out pois and where the concavity changes from concave up to concave down in this graph, i have to do the second derivative.

04:37 So let's get some space.

04:41 And we have to take our function and do the second derivative.

04:46 So i think the easiest way to do that would be to write that denominator, 3 plus 17 e to the negative 0 .0625 x and raise that denominator to the negative one power.

05:01 I think that's going to be the easiest way.

05:04 All right.

05:04 So here we go.

05:05 Uh -oh.

05:07 Lost my screen there.

05:08 The easiest way to – oh, i did it again – is y prime equals.

05:13 All right.

05:13 Bring your exponent down.

05:16 So that's going to be a negative 300.

05:18 Keep all of that.

05:21 That gets raised and subtract one from the exponent, so that becomes a negative 2.

05:30 Chain rule says you have to take the derivative of what's inside.