1. Functions

Lesson

We look for a model that will summarize an apparent relationship between two variable quantities. The model, if successful, will enable predictions to be made about further observations of an experiment and it may help in developing an understanding of the mechanisms involved in a process.

As a rule, we look for the simplest* *model that will provide a reasonably good fit for a set of data. And, of all possible models, a linear model is the simplest.

Even when it is clear that a nonlinear model would describe the data better than would a linear model, we may choose to use a linear model that will be approximately correct over a small portion of the data range. We may wish to approximate a complicated nonlinear process by a string of separate linear models that each apply to a small sub-range of the data.

It is proposed to carry out an experiment to do with trees that are being grown for timber in a plantation. The researchers are interested in the diameters of the tree trunks at time intervals after the trees were first planted. Before any measurements are made, various theories are discussed concerning the possible outcomes.

On the basis of this thinking, certain mathematical models are discarded as being impossible or unlikely and the experiment proceeds with the aim of discovering which of the remaining, plausible models seems to best reflect reality.

- The model would have to be an
*increasing*function, at least up to some point, because the trees will never reduce in size. - The rate of increase might decline with time because the available nutrients the tree needs for growth are going into increasing the height of the tree as well as the diameter.
- The rate of increase in diameter might decrease because this will still allow the cross-sectional area of a tree trunk to increase at a steady rate.
- The rate of increase in the diameter might stay fairly constant because the tree is developing a stronger root system to maintain its nutrient supply and its growth rate.
- The pattern of growth might be erratic and somewhat unpredictable because of variations in climate.

On the basis of this thinking, the researchers would probably discard an exponential model, a quadratic model, all periodic functions and various others, but might consider linear models (including composites of several linear pieces), a logarithm function, a square root function, or one of the several special S-shaped functions.

Every week, Rochelle deposits $\$800$$800 into a savings account. She plots her accumulated savings over several weeks.

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Would a linear model accurately describe Rochelle’s savings after the first week? Choose the most appropriate response.

No, Rochelle's savings will not change over time.

AYes, the difference in the balance from week to week increases at a linear rate.

B