A problem from Stewart *Calculus: Concepts and Contexts* 2/e p. 38. It appears in Chapter 1.

- Why use “estimate” rather than “interpolate”?
- What are the essential properties of the function needed to create a reasonable interpolation?
- What's the point of using a cubic?
- How would you decide whether to use a cubic or some other function?

`mosaic`

- Start R
- Load the package (you need do this only at the start of a session)

```
require(mosaic)
```

- Read in the data and display a bit

```
pop = fetchData("PREP-Stewart-World-Population.csv")
```

```
## Retrieving from
## http://www.mosaic-web.org/go/datasets/PREP-Stewart-World-Population.csv
```

```
head(pop)
```

```
## Year Population
## 1 1900 1650
## 2 1910 1750
## 3 1920 1860
## 4 1930 2070
## 5 1940 2300
## 6 1950 2560
```

What skills are needed to do this? Careful spelling, attention to punctuation, use of quotes, understanding the structure of a file name, understanding the syntax for use of R functions, understanding assignment.

- Plot the data

```
plotPoints(Population ~ Year, data = pop)
```

What do you need to know to answer the questions posed earlier?

Some techniques:

```
quadf = fitModel(Population ~ a * Year^2 + b * Year + c, data = pop)
quadf(1925)
```

```
## [1] 1878
```

```
quadfResids = with(pop, Population - quadf(Year))
```

You figure it out!

```
splinef = spliner(Population ~ Year, data = pop)
splinef(1925)
```

```
## [1] 1956
```

```
splinefResids = with(pop, Population - splinef(Year))
```

You can also try a monotonic spline. Use `help(spliner)`

to find out how.

This is hard, for reasons that relate to the data and numerics. Here, a guess is being made for a doubling time of 30 years.

```
expf = fitModel(Population ~ A + B * 2^((Year - 1900)/30), data = pop)
```

According to this model, what's the population in 1925?

- Why are there so many extra parameters in the functions? Why not just
`a*Year^3`

for the cubic? - Which function is right?
- How well do the various functions work for extrapolation? Look up the world population in 2010 and check. Also, look up the world population in 1500 and check.
- What factors might influence world population that might mean that the rules of growth in 2000 might be different than 1900? If the system is changing, why can a mathematical function that doesn't change in form over the years capture the dynamics of population?
- Suppose that your job is to predict the world population in 2020. How would you build a model for this purpose?