Math 214 - Multivariable Calculus - Prep

Task 20.1

Consider the function $f(x,y)=x^3-3x+y^2-4y$.

Find the critical points of $f$ by finding when $Df(x,y)$ is the zero matrix.
Find the eigenvalues of $D^2f$ at any critical points. [Hint: First compute $D^2f$. Since there are two critical points, evaluate the second derivative at each point to obtain 2 different matrices. Then find the eigenvalues of each matrix.]
Label each critical point as a local maximum, local minimum, or saddle point, and state the value of $f$ at the critical point.
Use Mathematica to construct a 2D contour plot and 3D surface plot of the function to visually verify that your solution is correct. Choose bounds for your plots so that the critical points are clearly visible.

The Mathematica Notebook 2ndDerTest.nb can help you check much of your work above.

Task 20.2

Let's now return to a Lagrange multiplier problem, where we have a constraint that limits the values over which we want to optimize a function. Consider the curve $xy^2=54$.

Start by drawing the curve.

The distance from each point on this curve to the origin is a function that must have a minimum value. We will find a point $(a,b)$ on the curve that is closest to the origin.

The first step to any Lagrange multiplier problem is to identify the function $f(x,y)$ that we wish to maximize or minimize, and then then identify the constraint and write it in the form $g(x,y) = c$. The distance from $(x,y)$ to the origin is $f(x,y) = \sqrt{(x-0)^2+(y-2)^2}=\sqrt{x^2+y^2}.$ This is the function we wish to minimize. The square root on this function will complicate computations later on. Because the square root function is increasing, note that $h(x,y) = x^2+y^2$ will have its minimum value at the same place. Because of this, we can simplify our work and use $f(x,y)=x^2+y^2$ as the function we wish to minimize.

What's the constant $c$ and function $g$ so that our constraint can be written in the form $g(x,y)=c$?
Solve the system $\vec \nabla f = \lambda \vec \nabla g$ and $g=c$.
- After computing the gradients, state the 3 equations that form the system we must solve, and then solve it.
- Note that in this problem, the number $\lambda$ is not an eigenvalue, rather it is a multiplier that helps us know if $\vec \nabla f$ and $\vec \nabla g$ lie on the same line (are parallel or antiparallel, i.e. "Is one gradient a multiple of the other?".
State the $(x,y)$ coordinates on the curve $xy^2=54$ that are closest to the origin.

Remember that you can use LagrangeMultipliers.nb to check your work.

How does the problem above change if we want to find the point on the curve that is closest to $(3,4)$? Solving the corresponding system of equations by hand will not be simple, but we can use the Mathematica notebook above to quickly answer this question, once we state $f$, $g$, and $c$. You will need to numerically approximate the solution that Mathematica gives (type //N at the end of a line of code to numerically approximate the output). The solution is $(x,y) = (3.11122,4.16612)$.

Task 20.3

Consider the function $f(x,y,z) = -x^2+y^2+z^2$.

Start by using the ContourPlot3D[] command in Mathematica to draw several level surface of this function. You can use the Mathematica notebook ContourSurfaceGradient.nb to help you.

The level surface which passes through the point $(3,2,-1)$ is shown below, along with the tangent plane to the surface through the point $(3,2,-1)$. This surface is called a hyperboloid of two sheets.

Use the differential $$df = f_xdx+f_ydy+f_zdz \quad\text{or}\quad df=\vec\nabla f(a,b,c)\cdot(dx,dy,dz) . $$ to give an equation of the tangent plane to this surface at the point $(3,2,-1)$. [Hint: Start by explaining why $df=0$. Then we have $dx=x-3$, $dy=y-?$, and $dz =?$. Don't forget to evaluate the partials at the correct point.]
Suppose the function $f(x,y,z) = -x^2+y^2+z^2$ gives the temperature (in Celcius) at points in space near some object (located at the origin), with $x,y,z$ values given in meters. Compute the temperature at $(3,2,-1)$, and then use differentials to approximate the temperature at $(3.01,1.98, -0.98)$. [What are $dx$, $dy$, and $dz$?]
Compute the directional derivative of $f(x,y,z) = -x^2+y^2+z^2$ at the point $(3,2,-1)$ in the direction $(1, -2, 2)$. What are the units of $D_{ (1, -2, 2) }f(3,2,-1)$?
What similarities, and what differences, do you see in the three questions above?

Task 20.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.

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Prep Day 20 Task 20.1 Consider the function $f(x,y)=x^3-3x+y^2-4y$. Find the critical points of $f$ by finding when $Df(x,y)$ is the zero matrix. Find the eigenvalues of $D^2f$ at any critical points. [Hint: First compute $D^2f$. Since there are two critical points, evaluate the second derivative at each point to obtain 2 different matrices. Then find the eigenvalues of each matrix.] Label each critical point as a local maximum, local minimum, or saddle point, and state the value of $f$ at the critical point. Use Mathematica to construct a 2D contour plot and 3D surface plot of the function to visually verify that your solution is correct. Choose bounds for your plots so that the critical points are clearly visible. The Mathematica Notebook 2ndDerTest.nb can help you check much of your work above. Task 20.2 Let's now return to a Lagrange multiplier problem, where we have a constraint that limits the values over which we want to optimize a function. Consider the curve $xy^2=54$. Start by drawing the curve. The distance from each point on this curve to the origin is a function that must have a minimum value. We will find a point $(a,b)$ on the curve that is closest to the origin. The first step to any Lagrange multiplier problem is to identify the function $f(x,y)$ that we wish to maximize or minimize, and then then identify the constraint and write it in the form $g(x,y) = c$. The distance from $(x,y)$ to the origin is $f(x,y) = \sqrt{(x-0)^2+(y-2)^2}=\sqrt{x^2+y^2}.$ This is the function we wish to minimize. The square root on this function will complicate computations later on. Because the square root function is increasing, note that $h(x,y) = x^2+y^2$ will have its minimum value at the same place. Because of this, we can simplify our work and use $f(x,y)=x^2+y^2$ as the function we wish to minimize. What's the constant $c$ and function $g$ so that our constraint can be written in the form $g(x,y)=c$? Solve the system $\vec \nabla f = \lambda \vec \nabla g$ and $g=c$. After computing the gradients, state the 3 equations that form the system we must solve, and then solve it. Note that in this problem, the number $\lambda$ is not an eigenvalue, rather it is a multiplier that helps us know if $\vec \nabla f$ and $\vec \nabla g$ lie on the same line (are parallel or antiparallel, i.e. "Is one gradient a multiple of the other?". State the $(x,y)$ coordinates on the curve $xy^2=54$ that are closest to the origin. Remember that you can use LagrangeMultipliers.nb to check your work. How does the problem above change if we want to find the point on the curve that is closest to $(3,4)$? Solving the corresponding system of equations by hand will not be simple, but we can use the Mathematica notebook above to quickly answer this question, once we state $f$, $g$, and $c$. You will need to numerically approximate the solution that Mathematica gives (type //N at the end of a line of code to numerically approximate the output). The solution is $(x,y) = (3.11122,4.16612)$. Task 20.3 Consider the function $f(x,y,z) = -x^2+y^2+z^2$. Start by using the ContourPlot3D[] command in Mathematica to draw several level surface of this function. You can use the Mathematica notebook ContourSurfaceGradient.nb to help you. The level surface which passes through the point $(3,2,-1)$ is shown below, along with the tangent plane to the surface through the point $(3,2,-1)$. This surface is called a hyperboloid of two sheets. Use the differential $$df = f_xdx+f_ydy+f_zdz \quad\text{or}\quad df=\vec\nabla f(a,b,c)\cdot(dx,dy,dz) . $$ to give an equation of the tangent plane to this surface at the point $(3,2,-1)$. [Hint: Start by explaining why $df=0$. Then we have $dx=x-3$, $dy=y-?$, and $dz =?$. Don't forget to evaluate the partials at the correct point.] Suppose the function $f(x,y,z) = -x^2+y^2+z^2$ gives the temperature (in Celcius) at points in space near some object (located at the origin), with $x,y,z$ values given in meters. Compute the temperature at $(3,2,-1)$, and then use differentials to approximate the temperature at $(3.01,1.98, -0.98)$. [What are $dx$, $dy$, and $dz$?] Compute the directional derivative of $f(x,y,z) = -x^2+y^2+z^2$ at the point $(3,2,-1)$ in the direction $(1, -2, 2)$. What are the units of $D_{ (1, -2, 2) }f(3,2,-1)$? What similarities, and what differences, do you see in the three questions above? Task 20.4 Pick some problems related to the topics we are discussing from the Text Book Practice page. Edit Page History Source Attach File Backlinks List Group Page last modified on October 12, 2022, at 05:16 PM
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Day 20

Task 20.1

Task 20.2

Task 20.3

Task 20.4