18.3 Nonlinear Models 599
Now that you realize the importance of polynomial models in describing engineering
situations, let us consider some of the basic characteristics of polynomial models. The general
form of a polynomial function ( model) is given by(18.9)
wherea 0 ,a 1 , ...,anare coefficients that could take on different values, andnis a positive integer
defining the order of the polynomial. For the laminar fluid velocity and the stopping sight distance
examples,nis 2, and the deflection of the beam was represented by a fourth-order polynomial.
Unlike linear models, second- and higher-order polynomials have variable slopes; mean-
ing each time you introduce a change in the value of the independent variablex, the cor-
responding change in the dependent variableywill depend on where in thexrange the change
is introduced. To better visualize the slope at a certainxvalue, draw a tangent line to the curve
at the correspondingxvalue, as shown in Figure 18.10. Another important characteristic of a
polynomial function is that the dependent variableyhas a zero value at points where it inter-
sects thexaxis. For example, for the laminar fluid velocity situation shown in Figure 18.6, the
dependent variable, velocityu, has zero values atr0.1 m andr0.1 m.
As another example, consider the third-order polynomialyf(x) x
3
6 x
2
3 x10,
as shown in Figure 18.10. This function intersects thexaxis atx1,x2, andx5. These
points are called thereal rootsof the polynomial function. Not all polynomial functions have
real roots. For example, the functionf(x) x
2
4 does not have a real root. As shown in
Figure 18.11, the function does not intersect thexaxis. This also should be obvious, because
if you were to solvef(x) x
2
4 0, you would findx
2
4. Even though this function
does not have a real root, it still possesses imaginary roots. You will learn about imaginary roots
in your advanced math and engineering classes.
Next, we consider other forms of nonlinear engineering models.yf 1 x 2 a 0 a 1 xa 2 x
2
a 3 x
3
pan x
n40
20
0
20
40
60
80
100
4 3 2 10f(x)
123 456
x
Slope
Slope
Slope
xf(x) x^3 6 x^2 3 x 10
3 80
2 28
10
010
18
20
3 8
4 10
50
628■Figure 18.10 The real roots of a third-order polynomial function.
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