1.2: Successive Differences and Degree | AP Precalculus

How it works

Finite differences compare outputs at equally spaced inputs. For a polynomial of degree n, the nth differences are constant and nonzero. A short data table can suggest a model, but other functions can interpolate the same finite set of points.

\Delta y_i=y_{i+1}-y_i,\quad\Delta^2y_i=\Delta y_{i+1}-\Delta y_i

Worked examples

Example 1

For inputs $\displaystyle 0,1,2,3$ , outputs are $\displaystyle 1,4,9,16$ . Find first and second differences.

First differences: $\displaystyle 3,5,7$ . Second differences: $\displaystyle 2,2$ . This is consistent with a quadratic model.

Example 2

Outputs at equally spaced inputs are $\displaystyle 2,5,8,11$ . Which simple model fits?

The first differences are all $\displaystyle 3$ , so a linear model fits; with inputs $\displaystyle 0,1,2,3$ , it is $\displaystyle y=2+3x$ .

Problems

Problem 1.2.1

Inputs $\displaystyle 0,1,2,3$ give outputs $\displaystyle 0,1,4,9$ . Find the first and second differences.

Show worked solution

First differences are $\displaystyle 1,3,5$ ; second differences are $\displaystyle 2,2$ . A quadratic model fits.

Problem 1.2.2

Can ordinary successive-difference degree tests be used unchanged at inputs $\displaystyle 0,1,3,6$ ?

Show worked solution

No. The input gaps are unequal. Equal-spacing assumptions fail; use divided differences or fit with the actual inputs.

Problem 1.2.3

Outputs $\displaystyle 1,8,27,64,125$ occur at inputs $\displaystyle 1,2,3,4,5$ . Find the third differences.

Show worked solution

First differences are $\displaystyle 7,19,37,61$ ; second differences are $\displaystyle 12,18,24$ ; third differences are $\displaystyle 6,6$ . This agrees with a cubic.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 1.2.1

Problem 1.2.2

Problem 1.2.3

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