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The table gives estimates of the world population, in millions, from 1750 to 2000.
(a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. Compare with the actual figures.(b) Use the exponential model and the population figures for 1850 and 1900 to predict the world population in 1950. Compare with the actual population.(c) Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 2000. Compare with the actual population and try to explain the discrepancy.
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Calculus 1 / AB
Exponential Growth and Decay
University of Nottingham
Idaho State University
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
The table gives estimates …
The table gives the popula…
World population The table…
The table gives the US pop…
Use the data in the table …
all right, We are given a table with various population values over time, and we have a lot of things to do. So I hope you packed a snack because this is going to take us a while. So in part A, we want to come up with an exponential model based on these two points from the table, and then we're going to use that model to come up with predictions for 1919 50. So to find the model, we first have to find the value of K we're going to use R P equals p 92 the Katie population growth equation. And we're going to use these two points. So we'll use for a final population 18 a 980 and we'll use for our initial population 790. And for time, what has elapsed between the year 17 50 the year 1800 is 50 years. So we use a 54 t, and we're going to solve this for K. So first we divide both sides by 7 90 it's going to reduce to 98 79th and then we're going to take the natural log of both sides and then we're going to divide by 50. So we get K is a natural log of 98 79th over 50 so we can substitute that into our model along with the initial population of 7 90 And for our model, we have p of tea equals 7 90 e to the power natural log 98 79th over 50 times T. Okay, so we're going to use that to make our prediction for the year 1900. So if it's 1900 then it has been 150 years since the year 17. 50. So this model assumes that t equals zero. Is the year 17 50? Okay, so for our prediction for the year 1900 we're going to use T equals 1 50 substitute that end to our population model, and we get approximately 1508. Now, that's 1508 million people. The actual value reading off the table was 16 50 16 50 million people. Okay, we're going to do the same thing for the year 1950. So if it's 1950 that it has been 200 years since the year 17 50 so we put 200 into our population model and we get approximately 18 71 18 71 million people, the actual model of the actual value from the table WAAS 2560. So notice that both of the values we got from this model were lower than the actual values. All right, that entire process, we're going to repeat for two different ordered pairs and make a different prediction. So for part B, this time we're using 18. 50 and its population of 12. 60 million and we're using 1900 and its population of 16. 50 1,000,000. We're going to go through the same process again. So we need to find the k value for this one using p of t equals P, not B to the K T. And then we'll have our model and then we can make our predictions. So we're substituting 16 50 for p of tea and we're substituting 12 64 p not. And this time again, 50 years has elapsed from 18 50 to 1900. So we have each of the 50 k. We're solving this for K. So we're going to start by dividing both sides by 12 60 that fraction will reduce to 55/42. 55/42 equals each of the 50 K. Then we take the natural log of both sides, and then we divide both sides by 50. So we have our K natural log of 55/42 divided by 50. Now, we substitute that into our model, and we have p of t equals 12 60. The population in the year 18 50 times e raised to the power natural log of 55/42 over 50 times. T. That's the model we're going to use for part B. Where t equals zero. Is the year 18 50. Okay, so the prediction we're making this time two of them, I believe, um or is it just one? Just one. We're making a prediction for the year 1950. So the year 1950 would be 100 years since the year 18 50. So we're going to use T equals 100. We substitute 100 into our model, and we get approximately 2161 million now, the actual value in that year based on our table was 25. 60 million. So another case of this model that we came up with giving us a value that's lower than the actual value. Now we have one more time to go. We're going to do Part C same idea. But this time the points were using are from 1900 with a population of 16 50 and from 1950 with a population of 25 60. Okay, let's use those values into our population growth equation and find the value of K. So the final value is 25 60 the initial value of 16 50 and the time that has elapsed is 50 years. Let's divide both sides by 16 50 on the fraction will reduce 2 to 56/1 65. Take the natural log on both sides and divide by 50. And that gives us our K value natural log of 2 56/1 65 all divided by K by 50. So we substitute that into our model P of t equals initial value 16 50 times. He raised to the power natural log of 2 56 over 1 65 over 50 times teeth and in this case, t equals zero is the year 1900 and the prediction were interested in making for part C is for the year 2000. So the year 2000 is going to be 100 years since the year 1900. So we're going to use T equals 100 substituted into our model. Put it in the calculator on would get approximately 3972 million for the population, while the actual population was significantly higher. 6080 million and that's over six billion people. So why the discrepancy? Well, if you think about what happened between 1919 50 and 2000 significant advances were made in medical technology in the last 50 years or so, and so life span would be incredibly altered. We couldn't use a model from the early 19 hundreds because of medical advances.
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