Preparation Problems#

Complete all preparation problems before class time on the scheduled date.

Week 15#

Day 15A: Monday, December 4th

No prep problem for today.

Day 15B: Wednesday, December 6th

No prep problem for today.

Day 15C: Friday, December 8th

Posted aftert Wednesday’s class.

Week 14#

Day 14A: Monday, November 27th

Evaluate the line integral where \(\int_{C}\vec{F}\cdot d\vec{r}\) where:

  • \(\vec{F}(x,y)=\langle -y, x \rangle \)

  • \(C\) is the curve with parametrization \(\vec{r}(t)=\langle 3\cos t, 3 \sin t \rangle \) and \(0 \leq t \leq\pi\)

Day 14B: Wednesday, November 29th

Evaluate the line integral where \(\int_{C}\vec{F}\cdot d\vec{r}\) where:

  • \(\vec{F}(x,y)=\langle 2xy^2, 2x^2y \rangle \)

  • \(C\) is the curve with parametrization \(\vec{r}(t)=\langle 3\cos t, 3 \sin t \rangle \) and \(0 \leq t \leq\pi /2\)

Day 14C: Friday, December 1st

No prep problem for today.

Week 13#

Day 13A: Monday, November 20th

Evaluate the following line integral where \(C\) is the line segment from \((2,1)\) to \((4,-3)\)

\[ \int_C 4y \; dx -2x \; dy \]
Day 13B: Wednesday, November 22nd

Thanksgiving Recess. No preparation for today.

Day 13C: Friday, November 24th

Thanksgiving Recess. No preparation for today.

Week 12#

Day 12A: Monday, November 13th

Plot the vectors for the vector plane \(\langle x+2, y \rangle \) at the points listed below:

  1. \((0,0)\)

  2. \((2,0)\)

  3. \((0,2)\)

  4. \((2,2)\)

Plot these on a single \(xy\)-plane. Use a straight edge to help with your sketch (neatness counts).

Day 12B: Wednesday, November 15th

Calculate the curl and the divergence of the vector field \(\vec{F}=\langle x^2, y^2, z^2 \rangle \)

Day 12C: Friday, November 17th

Find a potential function for the vector field:

\[ \vec{F} (x,y) = \lange x^2y, \tfrac{1}{3}x^3+4y \rangle \]

Week 11#

Day 11A: Monday, November 6th

No prep problem for today.

Day 11B: Wednesday, November 8th

Calculate the following triple integral by converting to cylindrical coordinates:

\[ \int_{-3}^{3} \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \int_{0}^{9-x^2-y^2} \left( x^2+y^2 \right) \; dz \; dy \; dx \]
Day 11C: Friday, November 10th

Convert the following points:

  1. Spherical to Cartesian: \(\left( 5, \dfrac{-\pi}{2}, \dfrac{\pi}{2} \right)\)

  2. Cartesian to Spherical: \(\left( -4, 0, 4 \right)\)

Week 10#

Day 10A: Monday, October 30th

No prep problem assigned for today.

Day 10B: Wednesday, November 1st

Find the volume of the solid bounded by the planes \(y=0\), \(z=0\), and the paraboloid \(z=4-x^2-y^2\).

Day 10C: Friday, November 3rd

Calculate the following triple integral where \(B=[1,2]\times[0,4]\times[0,1]\)

\[ \iiint_{B} xyz \; DV \]

Week 9#

Day 9A: Monday, October 23rd

Calculate the following double integral where \(D = \{ (x,y) | -2\leq x \leq 2, \quad -x^2\leq y \leq x^2+1 \}\)

\[ \int \int_{D} \left( x^2y+3 \right)\; dA \]
Day 9B: Wednesday, October 25th

Describe the following regions of the \(xy\)-plane as a Type 1 region using two inequalities. Sketch the region.

  1. Region bounded between \(x=0\), \(x=5\), \(y=4x\) and \(y=1-x^2\)

  2. Region bounded between \(y=x^2+1\) and \(y=5-x^2\)

  3. Region bounded by \(x^2+y^2=4\)

Day 9C: Friday, October 27th

No prep problem for today.

Week 8#

Day 8A: Monday, October 16th

Find and classify the critical point(s) of the following function:

\[ f(x,y)=x^2 - 6x + y^3 -12y^2 + 9 \]
Day 8B: Wednesday, October 18th

Find the extreme values of the function \(f(x,y)=4x^2+y^2\) on the circle \(x^2+y^2=1\).

Day 8C: Friday, October 20th

Calculate the iterated integral:

\[ \int_0^3 \int1_1^2 \left( 6x^2y+ 4x \right) dy dx \]

Week 7#

Day 7A: Monday, October 9th

No preparation problem assigned for today.

Day 7B: Wednesday, October 11th

Calculate the directional derivative of \(f(x,y)=5x^4-3xy^2\) where the unit vector is given by the following angles:

  1. \(\theta = \pi\)

  2. \(\theta = \pi/4\)

  3. \(\theta = \pi/3\)

Day 7C: Friday, October 13th

Find the critical point(s) for the following function:

\[ f(x,y)=\tfrac{1}{3}x^3 - 2x^2 + 3y^2 -12y + 9 \]

Week 6#

Day 6A: Monday, October 2nd

No preparation problem assigned for today.

Day 6B: Wednesday, October 4th

Find the equation of the plane tangent to the surface \(z=3x^2+4y^2\) at the point \(P(1,-1,7)\).

Day 6C: Friday, October 6th

Find \(\dfrac{dz}{dt}\) when \(t=0\) for:

\[ z=4x^3y-2xy^3 \]

where \(x=4\cos (2t)\) and \(y=4 \sin (2t)\).

Week 5#

Day 5A: Monday, September 25th

Find and sketch the domain of the following functions:

  1. \(f(x,y)=\sqrt{y-3x}\)

  2. \(g(x,y)=\dfrac{5}{y-x^2}\)

Day 5B: Wednesday, September 27th

Organize and bring all of your completed notes, problems, and papers with you to class.

Day 5C: Friday, September 29th

Study for Midterm Exam 1

Week 4#

Day 4A: Monday, September 18th

No problem assigned for today.

Day 4B: Wednesday, September 20th

Calculate the TNB-frame for the following curve:

\[ \vec{r}(t)=\langle 2+3\sin t,5,4+3\cos t \rangle \]
Day 4C: Friday, September 22nd

A moving particle starts at an initial position \(\vec{r}(0)=\lange 3,2,1\rangle\) and initial velocity \(\vec{v}(0)=\langle 0,1,-2\rangle\). Find the position function \(\vec{r}(t)\) given the acceleration:

\[ \vec{a}(t)=\langle 6e^t, 2\cos t , 4t \rangle \]

Week 3#

Day 3A: Monday, September 11th

V6: Find a linear equation for the plane that passes through the points \(P(2,0,1)\), \(Q(0,1,3)\), \(R(1,2,0)\).

Day 3B: Wednesday, September 13th

  1. Find the domain of the vector function \(\vec{r}(t)= \langle \sqrt{1-t^2}, t^2, \ln(t) \rangle\).

  2. Find the parametric equations for the line of intersection between the planes \(2x-y+3z=5\) and \(x+y-2z = 8\). (We did an example of this during class on Monday.)

Day 3C: Friday, September 15th

Find the arc length of the section of the circular helix given by

\[ \vec{r}(t)=\langle 2\sin t, 4t, 2\cos t \rangle \]

from the point \((0,0,2)\) to the point \((2,2\pi,0)\).

Week 2#

Day 2A: Monday, September 4th

No preparation problems assigned for today.

Day 2B: Wednesday, September 6th

  1. V4: Calculate the cross product \(\vec{a} \times \vec{b}\) of the following vectors: \( \vec{a}=\langle 2, 0, 1 \rangle \qquad \vec{b} = \langle 0, 2, 2 \rangle \)

  2. V4: Find a unit vector orthogonal to both of the following vectors: \( \vec{a}=\langle 1, 0, 3 \rangle \qquad \vec{b} = \langle -1, 2, 1 \rangle \)

Day 2C: Friday, September 8th

V5: Find parametric and symmetric equations for the line that passes through the points \(A(0,4,5)\) and \(B(-2,1,-1)\)

Week 1#

Day 1A: Monday, August 28th

No preparation problems assigned for today.

Day 1B: Wednesday, August 30th

Sketch the graph of the surface in \(\mathbf{R}^3\) given by the equation \(x=3\). On this graph, plot the points \((3,2,3)\) and \((3,2,-2)\)

Day 1C: Friday, September 1st

  1. V3: Calculate the dot product \(\vec{a} \cdot \vec{b}\) of the following vectors: \( \vec{a}=\langle 2, -3 \rangle \qquad \vec{b} = \langle 0, 4 \rangle \)

  2. V3: Calculate the dot product \(\vec{a} \cdot \vec{b}\) of the following vectors: \( \vec{a}=\langle 1, 9, 3 \rangle \qquad \vec{b} = \langle -4, 2, 3 \rangle \)

  • Show all work for these problems, do not just write the answer.