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.

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

Solve the systems algebraically.

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

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 $x_{1}.$

Express the area under the curve $y=x^{4}+5 x^{2}+x$ from 2 to 7 as a limit.

Find the volume of the cone. Round your answer to the nearest tenth. Radius = 2 ft, Height = 6 ft

Use the neoclassical model of investment to explain the impact of each of the following on the rental price of capital, the cost of capital, and investment. a. Anti-inflationary monetary policy raises the real interest rate. b. An earthquake destroys part of the capital stock. c. Immigration of foreign workers increase the size of the labor force. d. Advances in computer technology make production more efficient.

A small-appliance manufacturer finds that it costs $9000 to produce 1000 toaster ovens a week and$12,000 to produce 1500 toaster ovens a week. Express the cost as a function of the number of toaster