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'?

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.

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.

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

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

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.

Estimate the value of at which f increases most rapidly. Then find the exact value.

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

Solve the systems algebraically.

$$\left\{\begin{array}{l} {u+v=5} \\ {u-v=7} \end{array}\right.$$