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Section 8.5 Week 8 Activity

Subsection 8.5.1 Topology

Activity 8.5.1.

Consider the set \([0,5] \times [0,5] \times [0,5]\) in \(\RR^3\text{.}\) For each of the following points, determine if the point is a boundary point, an interior point, or neither.

  1. \(\displaystyle (3,3,3)\)

  2. \(\displaystyle (0,0,0)\)

  3. \(\displaystyle (0,5,0)\)

  4. \(\displaystyle (5,4,-1)\)

  5. \(\displaystyle (3,2,5)\)

Is the interval open, closed, or neither?

Solution.

  1. \((3,3,3)\) is an interior point.

  2. \((0,0,0)\) is a boundary point.

  3. \((0,5,0)\) is a bounardy point.

  4. \((5,4,-1)\) is a neither.

  5. \((3,2,5)\) is a boundary point.

The interval is closed, since it is formed from three closed intervals in \(\RR\text{.}\)

Activity 8.5.2.

Consider the set \((0,4) \times (2,4) \times (0,4)\) in \(\RR^3\text{.}\) For each of the following points, determine if the point is a boundary point, an interior point, or neither.

  1. \(\displaystyle (0,0,0)\)

  2. \(\displaystyle (4,4,4)\)

  3. \(\displaystyle (2,1,3)\)

  4. \(\displaystyle (0,2,3)\)

  5. \(\displaystyle (3,3,3)\)

Is the interval open, closed, or neither?

Solution.

  1. \((0,0,0)\) is neither. (It is outside the set, since it is well away from the \(y\) range.)

  2. \((4,4,4)\) is a boundary point.

  3. \((2,1,3)\) is neither (It is outside the set, since it is well away from the \(y\) range.

  4. \((0,2,3)\) is a boundary point.

  5. \((3,3,3)\) is an interior point.

The interval is open, since it is fromed from three open intervals in \(\RR\text{.}\)

Activity 8.5.3.

Consider the set \([1,3] \times (1,4) \times (0,4]\) in \(\RR^3\text{.}\) For each of the following points, determine if the point is a boundary point, an interior point, or neither.

  1. \(\displaystyle (2,2,2)\)

  2. \(\displaystyle (1,1,0)\)

  3. \(\displaystyle (3,2,2)\)

  4. \(\displaystyle (0,0,0)\)

  5. \(\displaystyle (2,1,0)\)

Is the interval open, closed, or neither?

Solution.

  1. \((2,2,2)\) is an interior point.

  2. \((1,1,0)\) is a boundary point.

  3. \((3,2,2)\) is a boundary point.

  4. \((0,0,0)\) is neither.

  5. \((2,1,0)\) is a boundary point.

The interval is neither open or closed. It includes is boundary planes in \(x\text{,}\) but not in \(y\) and only partially in \(z\text{.}\)

Subsection 8.5.2 Contour Plots

Activity 8.5.4.

Draw the contour graph for \(f(x,y) = 3y - 4x^2\) using a range of contour values \(c = -4, -3, \ldots, 5, 6\text{.}\) Use the contour graphs to give a qualitative description of the graph.

Solution.
Figure 8.5.1. Countour Plot for \(3y - 4x^2\)

The graph looks like a sloped ridge. The ridge goes upward along the positive \(y\) axis and slopes down to either side in the positive and negative \(x\) directions.

Activity 8.5.5.

Draw the contour graph for \(f(x,y) = e^{x^2 + y^2} + 4\) using a range of contour values \(c = 5, 5.1, 5.2, \ldots, 5.9, 6\text{.}\) Use the contour graphs to give a qualitative description of the graph.

Solution.
Figure 8.5.2. Countour Plot for \(e^{x^2 + y^2}\)

The graph looks like a well with a curve bottom. Near the origin, there is a slowly growing circular depression. As the function moves away from the origin, the shape remains circular but the walls get steep very quickly.

Activity 8.5.6.

Draw the contour graph for \(f(x,y) = \frac{x}{\sin y}\) using a range of contour values \(c = 0, 0.3, 0.6, \ldots, 2.4, 2.7, 3\text{.}\) Use the contour graphs to give a qualitative description of the graph.

Solution.
Figure 8.5.3. Countour Plot for \(\frac{x}{\sin y}\)

The graph is a series of ridges, alternating between ridges that grow in the positive \(x\) direction and ridges that grow in the negative \(x\) direction. The graph is undefined between the ridges due to the zeros of sine in the denominator.

Activity 8.5.7.

Draw the contour graph for \(f(x,y) = \frac{x^2 + 1}{y^2 - 4}\) using a range of contour values \(c = 1, 2, \ldots 9, 10\text{.}\) Use the contour graphs to give a qualitative description of the graph.

Solution.
Figure 8.5.4. Countour Plot for \(\frac{x^2+1}{y^2-4}\)

Near the origin, the graph is a hill-shape which cascades down quickly in all directions. Approaching the lines \(y = \pm 2\text{,}\) the graph descends to \(-\infty\text{.}\) Away from the origin, the graph has very steep walls growing to infinity near the lines \(y = \pm 2\text{,}\) but they shrink down to a flat plane as the input moves farther from the origin in the \(y\) direction.

Subsection 8.5.3 Conceptual Review Questions

  • Why are parametric curves a good mathematical tool for Kepler's Laws?

  • What is topology and why does it matter?

  • What are open and closed intervals in higher dimensions?

  • What is a multivariable function?

  • What is a countour plot?