courses:calculus_iii:fall_2013:daily_log

Class Log

Friday, 25 Oct

We discussed 8.1-8.5. For next time, please prepare 8.4 (which we discussed, but didn't do in class) and the optional problem after 8.5. Also, please do 8.6-8.9.

There is an online homework assignment over chapter 6 that is due next Friday, 1 Nov.

Friday, 04 Oct

The volume of a square-based pyramid with base side length $s$ and height $h$ $V(s,h)=s^2h/3$. Imagine that each of $V$, $s$, and $h$ depends on time $t$.

  1. If the height remains constant, but the radius changes, what is $dV/dt$ in terms of $ds/dt$? Use this to find a formula for $dV$ in terms of $ds$ when $h$ is constant.
  2. If the radius remains constant, but the height changes, what is $dV/dt$ in terms of $dh/dt$? What is $dV$ when $s$ is constant?
  3. If both the radius and height change, what is $dV/dt$ in terms of $dh/dt$ and $ds/dt$? Solve for $dV$.
  4. Show that we can write $dV$ as the matrix product of a 1-row by 2-column matrix with a 2-row by 1-column matrix: $$dV = \begin{bmatrix}? & ?\end{bmatrix}\begin{bmatrix}ds\\dh\end{bmatrix}.$$ How do the columns of the first matrix relate to the calculations you did above?

    The matrix $\begin{bmatrix}? & ?\end{bmatrix}$ is the derivative of $V$. The columns of this matrix are the partial derivatives of $V$.

  5. If we know that $s=3$ and $h=4$, and we know that $s$ increases by about $.1$ and $h$ decreases by about $.2$, then approximate how much $V$ will increase. Use your formula for $dV$ to approximate this.

Wednesday, 02 Oct

We started chapter 6, the derivatives chapter. We worked through problems 6.1-6.3 and talked about how the concepts behind the derivative problems. We also talked about epsilon-delta definitions of limits.

For Friday, please do problem 6.4, think about the challenge problem after 6.4, and do problems 6.5-6.8.

Wednesday, 25 Sep

We did some of these practice problems:

  1. Let $T(r,\theta)=(r\cos\theta, r\sin\theta)$ be the polar coordinate transformation. Draw the line $\theta=\pi/3$ for $-\infty\leq r\leq \infty$ (i.e., draw the curve $T(r,\pi/3)$). What is the Cartesian equation for this line (involving $x$ and $y$)?
  2. Consider all points that are a distance of 2 from the $z$ axis in $\mathbb{R}^3$. What does this look like? What is the Cartesian ($x,y$) equation for this shape?
  3. Define a transformation $T(\sigma,\tau)=(\sigma\tau,(1/2)(\tau^2-\sigma^2))$. Draw the lines $T(1,\tau)$, $T(2,\tau)$, $T(3,\tau)$ and $T(\sigma,1)$, $T(\sigma,2)$, $T(\sigma,3)$.

We talked about how to think about cylindrical and spherical coordinates, and went through problems 5.4 and 5.5. In the 11am class, we also talked about problem 4.16.

For Friday, please work through problem 5.10

Monday, 23 Sep

We discussed the last problem in chapter 4 (constructing the gravity vector field). We then did problems 5.1-5.3 in class. We spent some time talking about what a plane transformation is and visualizing the polar coordinate plane transformation.

For Wednesday, please work through problem 5.7. You may want to look in section 11.7 of the textbook for help in visualizing cylindrical or spherical coordinates.

Friday, 20 Sep

We discussed the rest of the problems in chapter 4. We also discussed applications of visualizing functions like vector fields, etc.

Over the weekend, please work on the two online homework assignments due next Friday (over parametric functions and representing functions). We will start the next chapter Monday.

Wednesday, 18 Sep

We did these practice problems:

  1. Describe the graph of $\vec r(a,b)=(a,b,a^2+3b^2)$ for $a,b\in [-3,3]$ (give a rough drawing, but also describe it in words and with your hands). If you have a real-world object you can shape into the graph, that's even better.
  2. Draw a rough contour plot of the function in the previous problem. Label the level curves with their height.

We then went through problems 4.1-4.6 or so. We continued to emphasize different ways to visualize information about a function (and especially what to do when we need more than 3 dimensions). We discussed parametric surfaces, and the wireframe you get when you hold each parameter constant and let the other vary (we see this in Problems 4.6 and 4.7 especially).

For Friday, please keep working through the problems, and try to work through Problem 4.16. Please make sure to click on the links to the side of the problems to check your work with a computer program. That will help you check your visualization skills.

I have also posted up two new online homework assignments over parametric functions and 3D functions. They are due a week from Friday, with the same 5-problem deal as before.

Monday, 16 Sep

We did this practice problem:

  1. For each of the following functions, determine the question marks in $f\colon \mathbb{R}^?\to\mathbb{R}^?$, draw a graph of the function, and tell how many dimensions is needed to fully represent the function:
    1. $f(t)=t^2$, $-1\leq t\leq 2$
    2. $f(t)=(t^2,t)$, $-1\leq t\leq 2$
    3. $f(t)=(t,\cos t, \sin t)$, $0\leq t\leq 4\pi$

      We spent a long time discussing how many dimensions is needed to fully represent a function (the sum of the domain and codomain dimensions), and how that information can be conveyed, especially when we have more than 3 dimensions to represent. We also discussed the “intrinsic” dimension of objects (e.g., a curve with one input is a 1-dimensional object, even if it lives in $\mathbb{R}^3$).

To prepare for the next problems, we then discussed how to represent functions like $f\colon \mathbb{R}^2\to \mathbb{R}^3$ (parametric surfaces) and $f\colon \mathbb{R}^2\to\mathbb{R}$ (using 3d plots and contour plots).

We didn't have time to hear any problems presented. For Wednesday, keep working through the problems and try to get through Problem 4.11.

Remember also that the Vectors assignment is due this Friday. You have 5 attempts for each problem. If you do not correctly answer the problem in 5 attempts, you can still get a perfect score, but you need to come talk with me (in office hours or set up an appointment) before the homework deadline. This means you shouldn't wait until Friday to do these problems, in case you have difficulty. I also opened back up the orientation problem set for a few more days.

Friday, 13 Sep

We did these practice problems:

  1. (the last practice problem from last time)
  2. Suppose a car's motion is described by $\vec r(t)=(3\cos(t), t^2+1)$. Write an integral calculating how far the car travels between $t=0$ and $t=5$.

Wednesday, 11 Sep

We did these practice problems:

  1. Let $\vec r(t)=(\cos t, 2\sin t)$ for $0\leq t\leq 2\pi$. Find the velocity, speed, and acceleration of the particle at $t=\pi/3$. Draw the vectors for velocity and acceleration, starting at the point $\vec r(\pi/3)$.
  2. Let $\vec r(t)=(\cos t, 2\sin t)$ for $0\leq t\leq 2\pi$. Find the slope $\dfrac{dy}{dx}$ and the concavity $\dfrac{d^2y}{dx^2}$ at $t=\pi/3, 2\pi/3, 4\pi/3, 5\pi/3$. Draw $\vec r(t)$ to visually check the slope and concavity at each point.

Please continue working on the problems in chapter 3. Try to finish the rest of them.

Monday, 9 Sep

We did these practice problems:

  1. $\vec F = c\vec d+\vec n$. Find $c$ and $\vec n$ in terms of $\vec F$ and $\vec d$ so that $\vec n \perp \vec d$.
  2. Graph $\vec r(t)=(\sin t, \cos t)$ for $0\leq t\leq 3\pi/2$. Show the start, end, and direction as well.

We worked through the last problems of chapter 2 and the first few problems of chapter 3 in the problem notes. For Wednesday, please work on the problems up through and including 3.10.

Friday, 6 Sep

We did this practice problem:

  1. Find the equation of a plane passing through the points $(2,1,1)$, $(1,0,1)$, and $(-3,0,1)$.

We then did presentations of problems, starting at problem 2.21.

4 Sep

We did these practice problems in class:

  1. Let $\vec F=(7,-4)$ and $\vec d=(2,1)$. Write $\vec F$ as the sum of a vector parallel to $\vec d$ and a vector orthogonal to $\vec d$. $\vec F=?+?$.
  2. Calculate $(3,2,1)\times (2,-1,1)$. Calculate the angle between your result and each of the vectors using the dot product.
  3. Find the equation of a plane passing through the point $(1,2,1)$ with normal vector $(2,-1,1)$. Is $(3,2,1)$ on the plane? Is $(-1/2,2,4)$ on the plane?

Then we had presentations and discussion about problems 2.17-2.20. We also had discussions comparing the cross and dot products.

For Friday, please continue working on the problems from chapter 2. Try to make it through Problem 2.30.

Aug 30

We did these practice problems in class:

  1. Calculate $(1,2,3)\cdot (-1,3,1)$
  2. Use the dot product formula $\vec u\cdot \vec v=|\vec u| |\vec v| \cos \theta$ to find the magnitude of $(1,2,3)$ by dotting the vector with itself.
  3. Use the dot product formula $\vec u\cdot \vec v=|\vec u| |\vec v| \cos \theta$ to find the angle between $(1,2,3)$ and $(-1,3,1)$. Are these vectors orthogonal? Make sure to label your answer as radians or degrees.
  4. Are $(4,2,1,3)$ and $(-1,2,3,-1)$ orthogonal?

We also talked about all vectors orthogonal to a given vector (how they form a line in 2d or a plane in 3d). We discussed why we should use radians instead of degrees in this class, and how the vector magnitude notation is a generalization of the absolute value notation. We also discussed questions that came up during the presentations of 2.10-2.16 and some of the big ideas to take away from 2.10-2.16 (for example, that the dot product gives us a way to calculate the length of a vector and the angle between two vectors, how the sign of the dot product helps us estimate the angle between two vectors, etc.)

Remember that we don't have class on Monday because of the holiday. For Wednesday, please work on problems 2.17-2.24. Some of these problems may be very familiar (we discussed some of them in class). You may find it helpful to watch several short videos I posted to Youtube:

Scanned copies of presented solutions are posted on Blackboard.

Aug 28

We did these practice problems:

  1. Let $\vec u=(2,3)$. Draw $\vec u$ with its tail at the origin. Calculate and draw $-3\vec u$. What does multiplying a vector by a constant do to the vector's arrow?
  2. Now copy down vector $\vec u$ on the board. Draw and label $2\vec u$ and $-3\vec u$.
  3. Let $\vec u=(2,3)$ and $\vec v=(-2,1)$. Draw $\vec u$, $\vec v$, and calculate and draw $\vec u+\vec v$. Calculate and draw $\vec u-\vec v$ and $\vec v-\vec u$.
  4. Now copy down $\vec u$ and $\vec v$ on the board. Draw and label $\vec u+\vec v$, $\vec u-\vec v$, and $\vec v-\vec u$.
  5. Find the magnitude of the vector $\vec v=(-1,1,2)$. Now multiply $\vec v$ by a number to get a unit vector (i.e., magnitude 1). What did you multiply by?
  6. Find the vector $\vec {PQ}$ that goes from $P=(0,1,2)$ to $Q=(-1,2,1)$. We'll use these points in the problems below too.
  7. Suppose a particle moves from $P$ to $Q$ (along $\vec {PQ}$) in 1 second. How fast is it going? What if it takes 2 seconds to move from $P$ to $Q$?
  8. Find a function $\vec r(t)=(?,?,?)t+(?,?,?)$ so that when $t=0$, $\vec r(0)=P$ and when $t=1$, $\vec r(1)=Q$. Check your work by plugging in $t=0$ and $t=1$.

After reviewing the main concepts from class practice problems, we had presentations of problems 2.4-2.9. Then we reviewed some definitions to prepare for 2.10-2.16.

For Friday, please work on solutions to 2.10-2.16. We noticed that the presentations go better if you write clearly and using a dark pen or something so that it shows up well on the document camera.

Aug 26

After introductions, we discussed problems 2.1-2.3 in the online notes (chapter 2, “Vectors, Lines, and Planes”). We started problem 2.4.

For Wednesday, please spend about two hours giving your best effort at working on problems 2.4-2.9. For each problem you think you have, write it up on a separate sheet of paper. We'll have people present solutions in class. If you present a solution in class, then you'll hand in your paper. I'll scan in and post the solutions that are presented in class.

I *highly* recommend working together to understand the problems, then writing up solutions individually.


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