Precalculus · Lesson 9.3

Technology and Interpretation

How it works

Technology can fit a model or approximate an intersection, but the result must still be interpreted with units, domain, and reasonable precision. A high fit statistic alone does not justify extrapolation far outside the data.

\text{residual}=\text{observed}-\text{predicted}

Worked examples

Example 1

A model predicts $\displaystyle 42$ while the observed value is $\displaystyle 45$ . Find the residual.

Residual is $\displaystyle 45-42=3$ . The model underpredicts by $\displaystyle 3$ units.

Example 2

A fitted cost model is $\displaystyle C(x)=12+3.5x$ . Interpret the parameters.

The fixed charge is $\displaystyle 12$ currency units and each additional input unit adds $\displaystyle 3.5$ currency units.

Problems

Problem 9.3.1

A linear model is $\displaystyle y=2x+1$ . Observations at $\displaystyle x=0,1,2$ are $\displaystyle 1,4,5$ . Find the residuals.

Show worked solution

Predictions are $\displaystyle 1,3,5$ , so residuals are $\displaystyle 0,1,0$ .

Problem 9.3.2

A calculator reports an intersection at $\displaystyle x=4.37291$ . A context asks for the nearest hundredth of an hour. Report the result.

Show worked solution

The time is approximately $\displaystyle 4.37$ hours. Keep extra digits while calculating and round the final result.

Problem 9.3.3

Why might a population model fitted over five years fail fifty years later?

Show worked solution

Resources, behavior, and growth rates may change. Extrapolation assumes the fitted pattern persists, which the original data cannot establish.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 9.3.1

Problem 9.3.2

Problem 9.3.3

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