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Chapter 9: Problem 13

By setting one variable constant, find a plane that intersects the graph of \(z=6 x^{2}-3 y^{2}+3\) in a: (a) Parabola opening upward: the plane ________=_________ (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (b) Parabola opening downward: the plane ________=_______ (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (c) Pair of intersecting straight lines: the plane _________=_________ (Give your answer by specifying the variable in the first answer blank and a value for it in the second.)

Short Answer

Expert verified
(a) Parabola opening upward: the plane \(y=k\) (b) Parabola opening downward: the plane \(x=k\) (c) Pair of intersecting straight lines: the plane \(z=k\)

Step by step solution

01

Parabola opening upward: the plane ________=_________

To construct a parabola opening upward, we'll set \(y\) to be a constant. So we write \(y=k\) where \(k\) is a constant. The function then becomes \(z=6x^2-3k^2+3\), is a parabola that opens upward, no matter what the value of \(k\) is. So to answer (a) we write \(y=k\).
02

Parabola opening downward: the plane ________=_______

Similarly, to construct a parabola opening downward, we'll set \(x\) to be a constant. So, we write \(x=k\), where \(k\) is another constant. The function then becomes \(z=-3y^2+3\), is a parabola that opens downward, with any value of \(k\). So to answer (b) we write \(x=k\).
03

Pair of intersecting straight lines: the plane _________=_________

For the graph to intersect in straight lines, we must set \(z = k\), where \(k\) is another constant. This simplifies the function to \(6x^2 - 3y^2 =k\). Depending on the value of \(k\), this equation could signify a couple of intersecting straight lines, therefore, to answer (c) we write \(z=k\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola
In mathematics, a parabola is a U-shaped curve that can open upwards or downwards. The general form of a parabola in a 3D setting is represented as a quadratic equation. In this exercise, we're dealing with the graph of the equation
  • \( z = 6x^2 - 3y^2 + 3 \)
To analyze parabolic sections here, you set one variable as a constant, effectively slicing or creating a cross-section of the three-dimensional shape. This technique turns the 3D graph into a 2D curve - the parabola. Let's explore both scenarios:- **Upward Opening**: By keeping \( y = k \) (where \( k \) is constant), the equation simplifies to \( z = 6x^2 - 3k^2 + 3 \), meaning it will graph as a parabola that opens upwards along the \( z \)-axis.- **Downward Opening**: When we set \( x = k \), the new equation is
  • \( z = -3y^2 + 3 \)
This indicates a parabola opening downwards, again viewed along the \( z \)-axis.Such simplifications reveal the nature of parabolas and how setting different variables as constants in multivariable calculus can alter their orientation.
Intersecting Lines
Intersecting lines are lines that cross at a particular point. In the realm of multivariable calculus and 3D plotting, it鈥檚 pivotal to understand how surfaces intersect with these lines. When we deal with an equation \( z = 6x^2 - 3y^2 + 3 \) and desire a pair of intersecting straight lines, we need to alter the equation by making
  • \( z = k \)
What we do here is essentially choosing a plane parallel to the \( xy \)-plane that intersects the surface. We arrive at the equation
  • \( 6x^2 - 3y^2 = k \)
Depending on the numeric value of\( k \), this equation can represent intersecting straight lines within that plane.
It's intriguing to note how different values of constant can adjust the equation to portray different intersection scenarios.
Planar Sections
Planar sections involve slicing a 3D object with a plane to explore different shapes and cross-sections within that object. In multivariable calculus, understanding planar sections is essential as it provides insight into the structure and behavior of 3D graphs.
Consider the surface described by
  • \( z = 6x^2 - 3y^2 + 3 \)
By setting either \( x \), \( y \), or \( z \) to a constant, you effectively create a planar section:
  • **Constant \( y \):** This creates an upward opening parabola.
  • **Constant \( x \):** This results in a downward opening parabola connected to the surface.
  • **Constant \( z \):** This planar section simplifies to a scenario where you can have intersecting lines.
Each of these setups gives a unique cross-section of the paraboloid. Understanding and visualizing these planar sections enable further study into how different dimensions interact within a graph.

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Most popular questions from this chapter

Find parametric equations for line that is tangent to the curve \(x=\) \(\cos t, y=\sin t, z=t\) at the point \(\left(\cos \left(\frac{5 \pi}{6}\right), \sin \left(\frac{5 \pi}{6}\right), \frac{5 \pi}{6}\right)\) Parametrize the line so that it passes through the given point at \(\mathrm{t}=0 .\) All three answers are required for credit. \(x(t)=\)_________ \(y(t)=\)_________ \(z(t)=\)_________

A bicycle pedal is pushed straight downwards by a foot with a 27 Newton force. The shaft of the pedal is \(20 \mathrm{~cm}\) long. If the shaft is \(\pi / 3\) radians past horizontal, what is the magnitude of the torque about the point where the shaft is attached to the bicycle? __________\(\mathrm{Nm}\)

Let \(a\) and \(b\) be positive real numbers. You have probably seen the equation \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\) that generates an ellipse, centered at \((h, k),\) with a horizontal axis of length \(2 a\) and a vertical axis of length \(2 b\). a. Explain why the vector function \(\mathbf{r}\) defined by \(\mathbf{r}(t)=\langle a \cos (t), b \sin (t)\rangle\), \(0 \leq t \leq 2 \pi\) is one parameterization of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) b. Find a parameterization of the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{16}=1\) that is traversed counterclockwise. c. Find a parameterization of the ellipse \(\frac{(x+3)^{2}}{4}+\frac{(y-2)^{2}}{9}=1\). d. Determine the \(x-y\) equation of the ellipse that is parameterized by $$ \mathbf{r}(t)=\langle 3+4 \sin (2 t), 1+3 \cos (2 t)\rangle $$

This exercise explores key relationships between a pair of planes. Consider the following two planes: one with scalar equation \(4 x-5 y+z=-2\), and the other which passes through the points \((1,1,1),(0,1,-1),\) and (4,2,-1) a. Find a vector normal to the first plane. b. Find a scalar equation for the second plane. c. Find the angle between the planes, where the angle between them is defined by the angle between their respective normal vectors. d. Find a point that lies on both planes. e. Since these two planes do not have parallel normal vectors, the planes must intersect, and thus must intersect in a line. Observe that the line of intersection lies in both planes, and thus the direction vector of the line must be perpendicular to each of the respective normal vectors of the two planes. Find a direction vector for the line of intersection for the two planes. f. Determine parametric equations for the line of intersection of the two planes.

Find the curvature of \(y=\sin (-2 x)\) at \(x=\frac{\pi}{4}\).
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