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Section 12.4 Assignment 4

  1. Assume that two particles have the following paths, both starting at \(t=0\text{.}\) Prove that the two particles collide, and find their speeds at the time of collision. (6)

    \begin{align*} \amp \gamma_1(t) = (t^2 + 6, 3t, t-2) \amp \amp \gamma_2(t) = (3t^2 - 4t - 10, 3t, t^2 - 4t + 2) \end{align*}

  2. For the following curve, calculate the vectors \(T(t), N(t)\) and \(B(t)\) as well as the speed, curvature and torsion. Use these to qualitatively describe the motion. (10)

    \begin{equation*} \gamma(t) = (e^t, e^t \sin t, e^t \cos t) \end{equation*}

  3. For the following curve, calculate the vectors \(T(t), N(t)\) and \(B(t)\) as well as the speed, curvature and torsion. Use these to qualitatively describe the motion. (10)

    \begin{equation*} \gamma(t) = (\sin t \cos t, \sin^2 t, 4t) \end{equation*}

  4. Choose a situation of a flycatcher chasing a mosquoto, a jet-fighter dogfight, or something similar with three dimensional motion. (Feel free to be creative in coming up with a scenario here.) Describe the motion of both parties in terms of curves with speed, curvature and torsion. (You don't need specific algebraic descriptions of specific curves. Instead, use speed, curvature and torsion to describe, qualitatively, what the curves would be doing. No calculations are required.) (4)

  5. Consider a function \(C(t,h,w,c)\) which describes human comfort in an environment where \(t\) is the temperature, \(h\) is the humidity, \(w\) is the windspeed and \(c\) is the percentage of the sky which is cloudy. (Feel free to make units where necessary.) Consider this a function of four variables, answer the following questions. (8)

    1. What is the domain of the function. (You should impose reasonable practical limits on domain, as well as mathematically necessary limits).

    2. What is the range of the function?

    3. Does the function increase or decrease in each of the variables?

  6. Consider a function \(p(t,o,l,s)\) measuring the population of a large fish species in an ecosystem, where \(t\) is the water temperature, \(o\) is oxygen concentration, \(l\) is the population of a little fish species which is a food source, and \(s\) is the population of a shark species which is a predator. Considering this as a function of four variables, answer the following questions. (8)

    1. What is the domain of the function. (You should impose reasonable practical limits on domain, as well as mathematically necessary limits).

    2. What is the range of the function?

    3. Does the function increase or decrease in each of the variables?

  7. Sketch the contour maps for this function and and describe the shapes of some of the level curves. (6)

    \begin{equation*} f(x,y) = x^2 - 2xy + y^2 \end{equation*}

  8. Sketch the contour maps for this function and and describe the shapes of some of the level curves. (6)

    \begin{equation*} f(x,y) = e^{\frac{y}{x}} \end{equation*}

  9. Which if the following sets are open, closed, both or neither? (Reasons are not required on this question.) (7)

    1. The empty set.

    2. The unit circle \(S^1\) in \(\RR^2\text{.}\)

    3. The set \(\{ (x,y,z) \in \RR^3 | x \lt y + z \}\text{.}\)

    4. The domain of the function \(f(x,y) = \frac{1}{y^2 - x^2}\text{.}\)

    5. The domain of the function \(f(x,y,z) = \ln(xyz)\text{.}\)

    6. The set of points in \(\RR^3\) which are less than three units from the \(y\) axis.

    7. \(\RR^3\) removing the unit sphere \(S^2\text{.}\)