Two vehicles start out traveling side by side along a straight road. Their position functions are given by s = f (t) and s = g(t), where s is measured in feet and t is measured in seconds. a. Which vehicle has traveled farther at t = 2 seconds? b. What is the approximate velocity of each vehicle at t = 3 seconds? c. Which vehicle is traveling faster at t = 4 seconds? d. What is true about the positions of the vehicles at t = 4 seconds?

(The QR Method) This problem outlines the basic ideas of an alternative method, the QR method, of finding the least squares estimate $\hat{\boldsymbol{\beta}}$. An advantage of the method is that it does not include forming the matrix $\mathbf{X}^{T} \mathbf{X}$, a process that tends to increase rounding error. The essential ingredient of the method is that if $\mathbf{X}_{n} \times p$ has p linearly independent columns, it may be factored in the form $\mathbf{X}=\mathbf{Q} \quad \mathbf{R}$ $n \times p \quad n \times p p \times p$ where the columns of Q are orthogonal $\left(\mathbf{Q}^{T} \mathbf{Q}=\mathbf{I}\right)$ and R is upper-triangular $\left(r_{i j}=0, \text { for } i>j\right)$ and nonsingular. [For a discussion of this decomposition and its relationship to the Gram-Schmidt process, see Strang (1980).] Show that $\hat{\boldsymbol{\beta}}=\left(\mathbf{X}^{T} \mathbf{X}\right)^{-1} \mathbf{X}^{T} \mathbf{Y}$ may also be expressed as $\hat{\boldsymbol{\beta}}=\mathbf{R}^{-1} \mathbf{Q}^{T} \mathbf{Y}$, or $\mathbf{R} \hat{\beta}=\mathbf{Q}^{T} \mathbf{Y}$. Indicate how this last equation may be solved for $\hat{\boldsymbol{\beta}}$ by back-substitution, using that R is upper-triangular, and show that it is thus unnecessary to invert R.

The active ingredient formed by aspirin in the body is salicylic acid, $\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{OH}\left(\mathrm{CO}_{2} \mathrm{H}\right)$. The carboxyl group ($-\mathrm{CO}_{2} \mathrm{H}$) acts as a weak acid. The phenol group (an OH group bonded to an aromatic ring) also acts as an acid but a much weaker acid. List, in order of descending concentration, all of the ionic and molecular species present in a 0.001-M aqueous solution of $\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{OH}\left(\mathrm{CO}_{2} \mathrm{H}\right)$.

Suppose C is a nonsingular irreducible affine curve over an algebraically closed field k. Prove that the coordinate ring k[C] is a Dedekind Domain.

Let

$$F_{n}=\mathbb{Z} G \otimes_{\mathbb{Z}} \mathbb{Z} G \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} \mathbb{Z}$$

G(n + 1 factors for n $\geq$ 0 with G-action defined on simple tensors by

$$g \cdot\left(g_{0} \otimes g_{1} \otimes \cdots \otimes g_{n}\right)=\left(g g_{0}\right) \otimes g_{1} \otimes \cdots \otimes g_{n}$$

(a) Prove that $F_{n}$ is a free $\mathbb{Z} G$-module of $\operatorname{rank}|G|^{n}$ with $\mathbb{Z} G$ basis

$$1 \otimes g_{1} \otimes g_{2} \otimes \cdots \otimes g_{n}$$

with $g_{i} \in G$. Denote the basis element

$$1 \otimes g_{1} \otimes g_{2} \otimes \cdots \otimes g_{n}$$

in (a) by $\left(g_{1}, g_{2}, \ldots, g_{n}\right)$ and define the G-module homomorphisms $d_{n}$ for n $\geq$ 1 on these basis elements by $d_{1}\left(g_{1}\right)=g_{1}-1$ and

$$\begin{array}{c} d_{n}\left(g_{1}, \ldots, g_{n}\right)=g_{1} \cdot\left(g_{2}, \ldots, g_{n}\right)+\sum_{i=1}^{n-1}(-1)^{i}\left(g_{1}, \ldots, g_{i-1}, g_{i} g_{i+1}, g_{i+2}, \ldots, g_{n}\right) \\ +(-1)^{n}\left(g_{1}, \ldots, g_{n-1}\right) \end{array}$$

for n $\geq 2$. Define the $\mathbb{Z}$-module contracting homomorphisms

$$\mathbb{Z} \stackrel{s_{-1}}{\rightarrow} F_{0} \stackrel{s_{0}}{\longrightarrow} F_{1} \stackrel{s_{1}}{\longrightarrow} F_{2} \stackrel{s_{2}}{\longrightarrow} \cdots$$

on a $\mathbb{Z}$ basis by $s_{-1}(1)=1$ and

$$s_{n}\left(g_{0} \otimes \cdots \otimes g_{n}\right)=1 \otimes g_{0} \otimes \ldots \otimes g_{n}.$$

(b) Prove that

$$\epsilon s_{-1}=1, \quad d_{1} s_{0}+s_{-1} \epsilon=1, \quad d_{n+1} s_{n}+s_{n-1} d_{n}=1, \text { for all } n \geq 1$$

where the map aug : $F_{0} \rightarrow \mathbb{Z}$ is the augmentation

$$\operatorname{map} \operatorname{aug}\left(\sum_{g \in G} \alpha_{g} g\right)=\sum_{g \in G} \alpha_{g}.$$

(c) Prove that the maps $s_{n}$ are a chain homotopy between the identity (chain) map and the zero (chain) map from the chain

$$\cdots \longrightarrow F_{n} \stackrel{d_{n}}{\longrightarrow} F_{n-1} \stackrel{d_{n-1}}{\longrightarrow} \cdots \stackrel{d_{1}}{\longrightarrow} F_{0} \stackrel{\text { aug }}{\longrightarrow} \mathbb{Z} \longrightarrow 0 \quad {(\ast)}$$

of $\mathbb{Z}$-modules to itself. (d) Deduce from (c) that all $\mathbb{Z}$-module homology groups of $(\ast)$ are zero, i.e., $(\ast)$ is an exact sequence of $\mathbb{Z}$-modules. Conclude that $(\ast)$ is a projective G-module resolution of $\mathbb{Z}$.

Let

$$u=\left[\begin{array}{l} 1 \\ 3 \end{array}\right]$$

and

$$v=\left[\begin{array}{r} 2 \\ -2 \end{array}\right],$$

and let A denote the point (-2, 1). a) Find points B and C such that $\mathbf{u}=\overrightarrow{A B}, \mathbf{v}=\overrightarrow{A C},$ and $\mathbf{u}-\mathbf{v}=\overrightarrow{C B}$. b) Graph $\mathbf{u}=\overrightarrow{A B}, \mathbf{v}=\overrightarrow{A C},$ and $\mathbf{u}-\mathbf{v}=\overrightarrow{C B}$

Why-other than the fact that the second law of thermodynamics says reversible engines are the most efficient-should heat engines employing reversible processes be more efficient than those employing irreversible processes? Consider that dissipative mechanisms are one cause of irreversibility.

Find the net rate of heat transfer by radiation from a skier standing in the shade, given the following. She is completely clothed in white (head to foot, including a ski mask), the clothes have an emissivity of 0.200 and a surface temperature of $10.0^{\circ} \mathrm{C},$ the surroundings are at $-15.0^{\circ} \mathrm{C},$ and her surface area is $1.60\ \mathrm{m}^{2}.$

(a) Prove that an arbitrary direct sum $\oplus_{i \in I} P_{i}$ of projective modules $P_{i}$ is projective and that an arbitrary direct product

$$\prod_{j \in J} Q_{j}$$

of injective modules $Q_{j}$ is injective. (b) Prove that an arbitrary direct sum of projective resolutions is again projective and use this to show

$$\operatorname{Ext}_{R}^{n}\left(\oplus_{i \in I} A_{i}, B\right) \cong \prod_{i \in I} \operatorname{Ext}_{R}^{n}\left(A_{i}, B\right)$$

for any collection of R-modules $A_{i}(i \in I)$. (c) Prove that an arbitrary direct product of injective resolutions is an injective resolution and use this to show

$$\operatorname{Ext}_{R}^{n}\left(A, \prod_{j \in J} B_{j}\right) \cong \prod_{j \in J} \operatorname{Ext}_{R}^{n}\left(A, B_{j}\right)$$

for any collection of R-modules $B_{j}(j \in J)$. (d) Prove that

$$\operatorname{Tor}_{n}^{R}\left(A, \oplus_{j \in J} B_{j}\right) \cong \oplus_{j \in J} \operatorname{Tor}_{n}^{R}\left(A, B_{j}\right)$$

for any collection of R-modules $B_{j}(j \in J).$

Prove the sums of squares identity for the two-way layout: $S S_{T O T}=S S_{A}+S S_{B}+S S_{A B}+S S_{E}$

Explain why it is not possible to prepare a ketone that contains only two carbon atoms.

Write Lewis structures for the cis–transisomers of $\mathrm{CH}_{3} \mathrm{CH}=\mathrm{CHCl}$.

A man consumes 3000 kcal of food in one day, converting most of it to maintain body temperature. If he loses half this energy by evaporating water (through breathing and sweating), how many kilograms of water evaporate?

Display all the possible configurations for a $(3 \times 2)$ matrix, $(3 \times 3)$ matrix, $(3 \times 4)$ matrix that is in echelon form. [Hint: There are seven such configurations. Consider the various positions that can be occupied by one, two, or none of the symbols.]