Find the area bounded by the line \(F(x)=2x+5\), the curve \(g(x)=x^2-8x+21\), and the two vertical lines \(x=2\) and \(x=5\).

Find the area of the region bounded by the two curves \(f(x)=x^2-6x+10\) and \(g(x)=-x^2+8x-10\), and the two vertical lines \(x=3\) and \(x=4\).

Find the area of the region that is completely bounded by the two curves \(f(x)=-x^2+4x+63\) and \(g(x)=x^2+12x+39\).

Find the area of the region that is completely bounded by the two curves \(f(x)=x^3-10x+25\) and \(g(x)=6x+25\).

The demand for a particular item is given by the function \(D(x)=1\text{,}450-3x^2\). Find the consumer’s surplus if the equilibrium price of a unit is \(\$250\).

Find both the consumer’s and the producer’s surplus for a product \(D(x)=25-0.005x^2\) and \(S(x)=0.004x^2\).

Revenue flows continuously into a retail operation at the constant rate of \(f(t)=128\text{,}000\) dollars per year. As the money is received, it is placed into a non-interest-earning account. Determine how much money has accumulated in the operation’s account at the end of a \(3\)-year time period.

Find the amount \(A\) of money that will have accumulated at the end of \(3\) years if a one-time, lump-sum investment of \(P=128\text{,}000\) dollars is made into an account paying \(7\%\) interest compounded continuously.

Suppose that at the end of each month, for \(6\) months, \(\$100\) is put into an account paying \(6\%\) annual interest compounded continuously. Except for the last \(\$100\), which earns no interest at all, each \(\$100\) earns interest over a different period of time. Using the continuous compounding formula \(A=Pe^{rt}\), find the total amount of money in the account at the end of \(6\) months.

How much money must be invested now at \(8\%\) interest compounded continuously, so that \(\$15\text{,}000\) will be available in \(10\) years?

Rental income from a piece of property, upon which there is an indefinite lease, flows at the rate of \(\$70\text{,}000\) per year. The income is invested immediately into an account paying \(6.5\%\) annual interest compounded continuously. Find and interpret the present value of the flow.

Solve \(\left(1+x^2\right)\dfrac{dy}{dx}=x\), and verify the solution.

Solve the differential equation \(y'=3x^2y^2\) subject to the condition that \(y=-0.1\) when \(x=2\).

Solve the differential equation \(y'=\dfrac{5x}{y^3}\) subject to the condition that \(y=3\) when \(x=-4\).

Solve the differential equation \(y'=x^2y\) subject to the condition that \(y=e^5\) when \(x=0\).

The value of real estate in a city is increasing, without limitations, at a rate proportional to its current value. A piece of property in the city appraised at \(\$240\text{,}000\) four months ago appraises at \(\$242\text{,}000\) today. At what value will the property be appraised \(12\) months from now?

The temperature of a machine when it is first shut down after operating is \(220^\circ\text{C}\). The surrounding air temperature is \(30^\circ\text{C}\). After \(20\) minutes, the temperature of the machine is \(160^\circ\text{C}\). Find a function that gives the temperature of the machine at any time, \(t\), and then find the temperature of the machine \(30\) after it is shut down.

Show that \[f(x)=\frac{3x^2}{124}\] is a valid probability density for each \(x\) in \([1, 6]\) given that \(f(x)=0\) for every \(x\) outside of \([1,5]\).

Find the probability that a randomly selected value of \(X\) lies in the interval \([1,3]\) if \[f(x)=\frac{3}{124}x^2\] Round the probability to \(4\) decimal places. (In Example 3 we showed that this function is a valid probability density function.)

A state’s weather service provides a continuously running \(120\)-second recorded telephone message about weather conditions throughout the state. If a person calls and gets connected to the message he or she hears the message at any point in the \(120\)-second run. If a person calls and gets connected, what is the probability that he or she will hear at most \(20\) seconds of the message before it repeats?

A machine is adjusted to dispense \(6\) ounces of coffee into a \(7\)-ounce cup. The amount of coffee dispensed is normally distributed mean \(\mu=6\) ounces and standard deviation \(\sigma=0.5\) ounces. Find the probability that a randomly selected cup is filled with between \(5.5\) and \(5.8\) ounces of coffee.

Find \(\dfrac{\partial f}{\partial x}\) and \(\dfrac{\partial f}{\partial y}\) for \(f(x, y)=8x^3+5y^2+6x-2y\) and evaluate each at \((2,7)\).

Find and interpret \(f_x(2, 3)\) and \(f_y(2, 3)\) for \[f(x, y)=x^4+8x^2y^3+5y^4\]

Find \(f_x\) and \(f_y\) for \(f(x, y)=\left(5x^3-8y^2\right)^4\)

Find \(f_x\), \(f_y\), and \(f_z\) for \(f(x, y, z)=e^{x+3z}+5\ln(xyz)\)

Find all the points \((x,y)\) for which both \(f_x\) and \(f_y\) equal zero, where \[f(x, y)=x^2+y^2-xy+y-8\]

Find the second partial derivatives of \(f(x, y)=x^3+2x^2y-5xy^2+3y^3\)