Recall that when dealing with integrals $$DfdV$$D$$fdV$,$ the function ff is called a density, or sometimes a potential when dealing with gradients F$=$fF$=$f$.$ This is because integrals like this often surround functions that represent things like density of an object or fluid, energy potentials, forces, etc.

Question $1\mathrm{.}1$ Sphere

Q$1\mathrm{.}1$ Sphere

$10$ Points

Grading comment:

Suppose the function f$($x$,$y$,$z$)=$x$2+$y$2$z$2$f$($x$,$y$,$z$)=$x$2+$y$2$z$2$ is the density of a solid at a point $($x$,$y$,$z$)($x$,$y$,$z$).$

Find the total mass of a hollow sphere of radius length $4.$ That is$,$ find:

M$=$Ef$($x$,$y$,$z$)$dVM$=$E$$f$($x$,$y$,$z$)$dV

where the sphere EE is the hollow sphere E$=\{($x$,$y$,$z$)$x$2+$y$2+$z$2=16\}$E$=\{($x$,$y$,$z$)$x$2+$y$2+$z$2=16\}.$

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Question $1\mathrm{.}1$: Sphere

Question $1\mathrm{.}2$ Maximum Density: Lagrange Multipliers

Q$1\mathrm{.}2$ Maximum Density: Lagrange Multipliers

$10$ Points

Grading comment:

Suppose the object in question is again a hollow sphere EE of radius $4,$ i$.$e$.$ x$2+$y$2+$z$2=16$x$2+$y$2+$z$2=16.$ Find the point$($s$)($x$,$y$,$z$)($x$,$y$,$z$)$ on the sphere that are the densest using Lagrange multipliers.

In other words: maximize f$($x$,$y$,$z$)=$x$2+$y$2$z$2$f$($x$,$y$,$z$)=$x$2+$y$2$z$2$ on the sphere x$2+$y$2+$z$2=16$x$2+$y$2+$z$2=16$ with Lagrange multipliers.

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Question $1\mathrm{.}2$: Maximum Density: Lagrange Multipliers

Question $1\mathrm{.}3$ Gradient Vector Field

Q$1\mathrm{.}3$ Gradient Vector Field

$10$ Points

Grading comment:

Compute the gradient vector field of the density function. That is$,$ find $$f$$f to measure how the density flows through the solid. What is the maximum rate of change at the top of the sphere, $(0,0,4)(0,0,4)?$

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Question $1\mathrm{.}3$: Gradient Vector Field

Question $1\mathrm{.}4$ Bonus: Maximum Change at Densest Point

Q$1\mathrm{.}4$ Bonus: Maximum Change at Densest Point

$5$ Points

Grading comment:

This problem is extra credit and is worth $5$ points on one of your past midterm exams $(10\%$ of them$).$ You can skip it with no consequences. If you earn credit via the successful completion of this problem, your total grade on any midterm cannot exceed $100\%.$

At one of the points of maximum density found in part $2$ of this problem, compute the direction of maximum rate of change. That is$,$ at the point where the sphere is densest, which direction is the density going to increase more if we could change the sphere?

For full credit, must use a point from part $2,$ and must include all computations. Final answer should be written as a sentence, e$.$g$.$ "the max rate of change at $($pt$)$ is in the direction $($direction$)"$ or something similar.