question id: not-a-mono-choice-1
12 Low-order polynomials
\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]
Reading questions
Reading question 12.1 “Poly” means many, as in “polygon” (having many sides) or “polymer” (a molecule formed by many identical chemical units) or “polygram” (a “lie detector” which records many physiological signals at the same time). In contrast, “mono” means one as in “monocle” (a visual aid for one eye) or “monopoly” (which comes from the Greek for “one seller.”)
- Provide a brief definition of “polynomial” in the mathematical sense. You definition should include the phrase “linear combination.”
- Say how a “monomial” differs from a polynomial.
Reading question 12.2
- Which one of the following is not a monomial?
- Which one of the following is is a monomial?
question id: not-a-mono-choice-2
- Which one of the following is is a polynomial?
question id: not-a-mono-choice-3
Reading question 12.3
- All polynomials are linear combinations of functions. The “linear polynomial” refers to a specific set of functions. Which are they?
question id: linear-polynomial-1
- Which of the panels in XREF not implemented yet show linear polynomials?
question id: linear-polynomial-2
- Explain why \(f(x) \equiv A (x - x_0)\) is a linear polynomial.
- Explain why \(g(t) \equiv A (x - x_0)^2\) is not a monomial.
Reading question 12.4 What makes a polynomial term an “interaction” term?
You can adjust the coefficient on it interactively.
It involves two monomials, each in a different input, being multiplied together.
It comes in the middle of a polynomial.
It has a coefficient of 1.
question id: interaction-term
Reading question 12.5 In the “low-order” polynomials described in the reading, what terms have the highest exponents?
linear terms
interaction terms
quadratic terms
cubic terms
question id: highest-order
Reading question 12.6 Which of the following is not a shape seen in a graph of a low-order polynomial in two variable?
question id: 2-var-shape
In the polynomial \[a_0 + a_x x + a_y y + a_{xx} xx\], what is the coefficient on the interaction term?
question id: coef-interaction
Reading question 12.7 Imagine a second-order polynomial in three inputs: \(x\), \(y\), and \(z\), like this: \[b_0 + b_x x + b_y y + b_z z + b_{xy} xy + b_{xz} xz + b_{xx} x^2 + b_{yy} y^2 + b_zz z^2\ .\] All of the possible second-order (or less) terms are shown, except for one. Which term is missing?
the constant term
the quadratic term in \(y\)
the interaction between \(z\) and \(x\)
the linear term in \(z\)
question id: missing-term