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Applied Calculus
Applications of Integration

1
Area of Regions in the Plane

Problem  1
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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 \(x=5\).

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Nathan
Nathan
Octabio cc
Octabio
Octabio cc espanol spanish
Octabio
Problem  2
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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 \(x=4\).

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Nathan
Nathan
Breylor cc
Breylor
Saba cc
Saba
Problem  3
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Find the area of the region that is completely bounded by the two curves \(f(x)=-x^2+4x+63\) and \(g(x)=x^2+12x+39\).

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Spencer
Spencer
Nathan
Nathan
Problem  4
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Find the area of the region that is completely bounded by the two curves \(f(x)=x^3-10x+25\) and \(g(x)=6x+25\).

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Saba cc
Saba
Spencer
Spencer
Nathan
Nathan

2
Consumer's and Producer's Surplus

Problem  1
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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 \(\$250\).

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

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Saba cc
Saba
Nathan
Nathan
Breylor
Breylor
Mr. Burzynski
Mr. Burzynski
Problem  3

The demand and supply for a particular type of chemical glassware is modeled by \[D(x)=-0.001x^2+154\] and \[S(x)=0.0002x^2+0.03x+100\] respectively, where \(x\) is the number of glassware units (in thousands) demanded and supplied, and both \(D(x)\) and \(S(x)\) are the glassware unit prices in dollars.

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Saba cc
Saba
Nathan
Nathan
Breylor
Breylor

3
Annuities and Money Streams

Problem  1
practice icon

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.

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Saba cc
Saba
Nathan
Nathan
Breylor
Breylor
Problem  2
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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.

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Nathan
Nathan
Breylor
Breylor
Problem  3
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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.

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Saba cc
Saba
Nathan
Nathan
Breylor
Breylor
Problem  4

Approximate the future value of the annuity in Example 3 over

  1. a \(6\)-month period

  2. a \(20\)-year period

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Nathan
Nathan
Breylor
Breylor
Problem  5
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How much money must be invested now at \(8\%\) interest compounded continuously, so that \(\$15\text{,}000\) will be available in \(10\) years?

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Saba cc
Saba
Octabio cc
Octabio
Nathan
Nathan
Octabio cc espanol spanish
Octabio
Problem  6
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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.

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Breylor
Breylor
Nathan
Nathan

4
Differential Equations

Problem  1
  1. Show that the differential equation \[\frac{dy}{dx}=15x^2y\] is separable.

  2. Verify, by differentiation, that the function \[y=Ce^{5x^3}\] is a solution to the separable differential equation \[\frac{dy}{dx}=15x^2y\]

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

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

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

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Octabio cc
Octabio
Nathan
Nathan
Octabio cc espanol spanish
Octabio
Problem  5
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Solve the differential equation \(y'=x^2y\) subject to the condition that \(y=e^5\) when \(x=0\).

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Saba cc
Saba
Spencer
Spencer
Nathan
Nathan

5
Applications of Differential Equations

Problem  1
practice icon

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?

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Spencer
Spencer
Nathan
Nathan
Problem  2
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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.

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Octabio cc
Octabio
Spencer
Spencer
Octabio cc espanol spanish
Octabio

6
Probability

Problem  1
practice icon

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 \([1,5]\).

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Octabio cc
Octabio
Nathan
Nathan
Octabio cc espanol spanish
Octabio
Problem  2
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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.)

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Nathan
Nathan
Problem  3
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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?

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Nathan
Nathan
Problem  4
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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.

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Nathan
Nathan

7
Partial Derivatives

Problem  1

The following examples will illustrate the meaning of \[\frac{\partial}{\partial y}f(x,y)=14x+5y+12\] for several different values of \(x\) and \(y\).

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Saba cc
Saba
Nathan
Nathan
Breylor
Breylor
Problem  2
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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 \((2,7)\).

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Octabio cc
Octabio
Nathan
Nathan
Octabio cc espanol spanish
Octabio
Problem  3
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Find and interpret \(f_x(2, 3)\) and \(f_y(2, 3)\) for \[f(x, y)=x^4+8x^2y^3+5y^4\]

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

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

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Nathan
Nathan
Saba cc
Saba
Breylor
Breylor
Problem  6
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Find all the points \((x,y)\) for which both \(f_x\) and \(f_y\) equal zero, where \[f(x, y)=x^2+y^2-xy+y-8\]

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Octabio cc
Octabio
Nathan
Nathan
Octabio cc espanol spanish
Octabio
Problem  7
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Find the second partial derivatives of \(f(x, y)=x^3+2x^2y-5xy^2+3y^3\)

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Nathan
Nathan
Saba cc
Saba
Breylor
Breylor
Problem  8

This is the solution to Problem 9 in Problem Set 7.2:

For \(f(x, y)=\dfrac{x^2+3y}{5x-3y^2}\), find

  1. \(f_x\)

  2. \(f_y\)

  3. \(f_x(0, 2)\)

  4. \(f_y(0. 1)\)

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Mr. McKeague
Mr. McKeague
Mr. Galletti
Mr. Galletti

8
Applications of Integration

No examples are currently assigned to this topic.