Let V be an inner product space with a subspace W having $B=\left\{\mathbf{w}_{1}, \mathbf{w}_{2}, \dots, \mathbf{w}_{n}\right\}$ as an orthonormal basis. Show that the function $T: V \rightarrow W$ represented by $T(\mathbf{v})=\left\langle\mathbf{v}, \mathbf{w}_{1}\right\rangle \mathbf{w}_{1}+\left\langle\mathbf{v}, \mathbf{w}_{2}\right\rangle \mathbf{w}_{2}+\cdots+\left\langle\mathbf{v}, \mathbf{w}_{n}\right\rangle \mathbf{w}_{n}$ is a linear transformation. T is called the orthogonal projection of V onto W.

How can you figure out the size of a part of a part without having to draw a diagram? Work with your team or your class to explore this question as you consider the example of $\frac{2}{3} \cdot \frac{4}{5}.$ a. Describe how you could draw a diagram to make this calculation. b. If you completed the diagram, how many parts would there be in all? How do you know? c. How many of the parts would be counted for the numerator of your result? Again, describe how you know. d. How can you know what the numerator and denominator of a product will be without having to draw or envision a diagram each time? Discuss this with your team and be prepared to explain your ideas to the class.

If we perform a single row operation on an identity matrix. we obtain an elementary matrix $\mathbf{E}_{\mathrm{Int}}, \mathbf{E}_{\mathrm{Repl}}, \text { or } \mathbf{E}_{\mathrm{Scale}}$ Find the elementary matrices for each of the following row operations on $I_3.$ (a) Interchange rows 1 and 2 $\left(\mathbf{E}_{\ln t}\right)$ (b) Add k times row 1 to row 3 $\left(\mathbf{E}_{\mathrm{Repl}}\right)$ (c) Multiply k times row $2\left(\mathbf{E}_{\text {Scale }}\right).$

Find the marginal cost for producing x units. (The cost is measured in dollars.) C=2500+320x

To enclose a yard, a fence is built against a large building, so that fencing material is used only on three sides. Material for the ends costs $15 per ft; material for the side opposite the building costs$25 per ft. Find the dimensions of the yard of maximum area that can be enclosed for $2400.

Kelly and friends buy a house after graduating from college and borrow $200,000 from the bank to pay for it. Suppose that the bank charges 12$ 2,500 to the bank. Call A(t) the amount of money the group still owes the bank. (a) What is the initial value problem that describes A(t)? (b) Solve the IVP in part (a). (c) How long will it take Kelly and friends to pay off the loan'?

Each of the following aldehydes or ketones is known by a common name. Its substitutive IUPAC name is provided in parentheses. Write a structural formula for each one. (a) Chloral (2,2,2-trichloroethanal) (b) Pivaldehyde (2,2-dimethylpropanal) (c) Acrolein (2-propenal) (d) Crotonaldehyde [(E)-2-butenal] (e) Citral [(E)-3,7-dimethyl-2,6-octadienal] (f) Diacetone alcohol ( 4-hydroxy-4- methyl-2-pentanone) (g) Carvone ( 5-isopropenyl-2-methyl-2-cyclohexenone) (h) Biacetyl (2,3-butanedione)

Compound A has the molecular formula $\mathrm{C}_{14} \mathrm{H}_{25} \mathrm{Br}$ and was obtained by reaction of sodium acetylide with 1, 12-dibromododecane. On treatment of compound A with sodium amide, it was converted to compound B ($\mathrm{C}_{14} \mathrm{H}_{24}$). Ozonolysis of compound B gave the diacid $\mathrm{HO}_{2} \mathrm{C}\left(\mathrm{CH}_{2}\right)_{12} \mathrm{CO}_{2} \mathrm{H}$. Catalytic hydrogenation of compound B over Lindlar palladium gave compound C ($\mathrm{C}_{14} \mathrm{H}_{26}$), and hydrogenation over platinum gave compound D ($\mathrm{C}_{14} \mathrm{H}_{28}$). C yielded $\mathrm{O} \square \mathrm{CH}\left(\mathrm{CH}_{2}\right)_{12} \mathrm{CH} \square \mathrm{O}$ on ozonolysis. Assign structures to compounds A through D so as to be consistent with the observed transformations.

Find the following. $\int\left(6 t^{2}-8 t+7\right) d t$

Determine concavity and the x-values where points of inflection occur. Do not sketch the graphs. $y=\frac{1}{10} x^{5}-3 x^{3}+17 x+43$

Show that the function $f(x)=\frac{x}{x+1}$ is increasing on any interval not containing x=-1.

Based on the E1 mechanism for the conversion of a tertiary alkyl chloride to an alkene as summarized in arrows 3 and 4, which arrow(s) correspond(s) to exothermic processes? A. Arrow 3 B. Arrow 4 C. Both 3 and 4 D. Neither 3 nor 4

In a discussion of an inferior good, Persky$^{4}$ solves an equation of the form $u_{0}=A \ln \left(x_{1}\right)+\frac{x_{2}^{2}}{2}$ for $x_{1},$ where $x_{1}$ and $x_{2}$ are quantities of two products, $u_{0}$ is a measure of utility, and A is a positive constant. Determine $x_{1}.$

Solve the systems algebraically.

$$\left\{\begin{array}{l} {\frac{1}{2} z-\frac{1}{4} w=\frac{1}{6}} \\ {\frac{1}{2} z+\frac{1}{4} w=\frac{1}{6}} \end{array}\right.$$