Find the indefinite integral of the function: f(x) = x^2(cos(x)) dx.
This problem asks us to integrate a function which is comprised of two differing types of functions. This requires us to use integration by parts which is: Integral(UDV) dx = UV - Integral(VDU) dx Here you are going to want to choose one of the variables to be U and the other to be DV. It is best to choose a U that will overtime differentiate down to a constant. In this case you would want to choose U = x^2, because if you differentiate twice the x^2 will eventually equal out to a constant. Once you have chosen your variables you want to differentiate U and integrate DV to get the values for DU and V. U = X^2 DV = Cos(x) DU = 2x V = Sin(x) Once you have these values you plug them into the equation and repeat the steps if necessary. Integral(x^2cos(x)) = x^2sin(x) - Integral(2xsin(x)) In this problem we are going to have to apply the parts forumla one last time to get rid of the 2x. U = 2x DV = sin(x) DU = 2 V = -cos(x) Integral(x^2cos(x)) = x^2sin(x) - [ -2xcos(x) - integral(-2cos(x) ] Now the integral is easy to compute and wont cause us trouble. Integral(x^2cos(x)) = x^2sin(x) - [ -2xcos(x) - (-2sin(x))] Simplifying: Integral(x^2cos(x) = x^2sin(x) + 2xcos(x) -2sin(x) + C Don't forget to add the constant of integration!
Why is the Marginal Cost Curve equivalent to the supply curve of a firm?
The Marginal Cost curve tells you the cost that you incur from each additional unit. Overtime, due to the law of diminishing marginal returns, the cost to produce each additional unit will increase. When matched with the Marginal Revenue curve, which shows you the revenue you will receive from producing an additional unit, it becomes clear the point at which you will want to produce. That is at the intersection between Marginal Revenue and Marginal Cost because at that point if you were to produce one more unit, your Marginal Cost would exceed your Marginal Revenue, meaning you would be losing money by producing another unit. This relationship effectively tells you how many units you will be producing just based on the Marginal Cost Curve.
List and define all the tools the Federal Reserve can use to control the money supply.
Open Market Operations - The Federal Reserve buys and sells government backed securities in the financial markets to control the money supply. Whether the money supply grows or shrinks depends on what operation the fund undertakes. If the Fed decides to sell securities it shrinks the money supply because the market will buy the security giving money to the Fed. If the Fed decides to buy securities then the money supply will increase because the Fed in paying money to the firms selling the securities. Discount Rate - The discount rate is the interest rate that is placed on overnight loans to commercial banks and other financial institutions from the Fed itself. While this is mostly used to add liquidity to the banking system, it can be used to help shrink or grow the money supply. If the rate is low, banks would be tempted to increase their reserves by borrowing, which will allow the banks to loan out more money increasing the money supply. If the interest rate is high, this will dissuade banks from borrowing money from the Fed and relying on keeping higher reserves which will decrease the amount banks can loan out which shrinks the money supply. Reserve Ratio - This tool is rarely used because of the huge impact it has on the money supply. The Fed has the ability to set the required reserves a bank must have in its vault. As such, all the reserves that exceed the required reserves can be turned into a loan. So if the Fed sets the reserve ratio high, that decreases the amount of money that can be loaned by the bank, which decreases the money supply. If the ratio is set low, that allows for banks to have a high amount of money that can be loaned, which increases the money supply. Interest rates on money deposits at the Fed - Recently a new tool has been introduced and that is the Interest that the Fed pays on bank deposits at the Fed. Currently, banks are allowed to deposit money into the Federal Reserve, think of this as like a savings account. The Fed pays interest on this money deposited which incentives banks to keep money in the accounts. The interest rate on this deposit is what incentives the bank to either keep money in the Fed or withdraw and keep it in its own vault or loan it out. If the interest rate is high, banks will want to deposit their money in the Fed, shrinking the money supply. If the rate is low, banks will want to withdraw their money and invest it in other areas, which would increase the money supply.