/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 Verify that the curve \(\mathbf{... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 52

Verify that the curve \(\mathbf{r}(t)\) lies on the given surface. Give the name of the surface. $$\mathbf{r}(t)=(t \cos t) \mathbf{i}+(t \sin t) \mathbf{j}+t \mathbf{k} ; x^{2}+y^{2}=z^{2}$$

Short Answer

Expert verified
Question: Prove that the curve defined by the vector function \(\mathbf{r}(t) = (t \cos t) \mathbf{i} + (t \sin t) \mathbf{j} + t \mathbf{k}\) lies on the surface defined by the equation \(x^{2} + y^{2} = z^{2}\) and identify the name of the surface. Solution: We substituted the components of the vector function into the surface equation and simplified it. The resulting equation holds for all valid values of \(t\), proving that the curve lies on the surface. The surface is a cone in a 3D space.

Step by step solution

01

Extract components from the given vector function

We have the vector function \(\mathbf{r}(t) = (t \cos t) \mathbf{i} + (t \sin t) \mathbf{j} + t \mathbf{k}\). We can extract its components as follows: $$x(t) = t \cos t$$ $$y(t) = t \sin t$$ $$z(t) = t$$
02

Substitute the components into the equation of the surface

Now we substitute these components into the surface equation: $$x^{2} + y^{2} = z^{2} \implies (t \cos t)^{2} + (t \sin t)^{2} = (t)^{2}$$
03

Simplify the equation and check if it holds for all values of \(t\)

Let's simplify and check whether the equality holds for all values of \(t\): $$t^{2} \cos^{2} t + t^{2} \sin^{2} t = t^{2}$$ Factor out \(t^{2}\) from the left side: $$t^{2} (\cos^{2} t + \sin^{2} t) = t^{2}$$ We know that \(\cos^{2} t + \sin^{2} t = 1\) (trigonometric identity). So we have: $$t^{2} (1) = t^{2}$$ This equation holds for all valid values of \(t\), hence the curve lies on the surface.
04

Name of the surface

The given surface equation is \(x^{2} + y^{2} = z^{2}\). This is the equation of a cone in a 3D space. Therefore, the name of the surface is a cone.

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Key Concepts

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

Vector Functions
A vector function is a crucial tool in representing curves in space. It is typically written in the form \( \mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} + z(t) \mathbf{k} \), where \(t\) is a parameter, and \(x(t)\), \(y(t)\), and \(z(t)\) are functions that describe the coordinates of a point along the curve as \(t\) varies.
For example, with the vector function \( \mathbf{r}(t) = (t \cos t) \mathbf{i} + (t \sin t) \mathbf{j} + t \mathbf{k} \), we see a vivid description of movement in 3D space:
  • \(x(t) = t \cos t\): Horizontal movement depending on trigonometric cosine function.
  • \(y(t) = t \sin t\): Vertical movement based on sine function.
  • \(z(t) = t\): Linear progression along the z-axis.
Vector functions help visualize and analyze the motion along a path, capturing complex 3D trajectories.
Trigonometric Identities
Trigonometric identities simplify mathematical expressions by revealing relationships among trigonometric functions. The fundamental identity \( \cos^2 t + \sin^2 t = 1 \) plays a critical role in various applications, including the analysis of curves.
Here鈥檚 why it鈥檚 essential:
  • When used in vector functions, it helps verify the relationships among components, assisting in proving properties like lying on a surface.
  • It allows for simplification, resulting in elegant solutions that are often less computationally demanding.
In the given exercise, leveraging \( \cos^2 t + \sin^2 t = 1 \) helped confirm that the equation \( x^2 + y^2 = z^2 \) holds true for all \(t\), affirming the curve resides precisely on the surface identified as a cone.
3D Surfaces
Understanding 3D surfaces like cones, spheres, and cylinders is vital in grasping spatial relationships. Surfaces in three dimensions are characterized by mathematical equations tying together three variables: \(x, y, \) and \(z\).
A surface like \( x^2 + y^2 = z^2 \) defines a geometric shape filling the 3D space, in this case, a cone. Key aspects:
  • 3D surfaces enable visual representation of complex equations and predict motion paths.
  • Equations give explicit relationships, showing how variables transform: from circles, ellipses to cones, and complex shapes.
The equation \( x^2 + y^2 = z^2 \) particularly simplifies to a double-napped cone when extended in 3D, providing insights into real-world shapes encountered in architecture and design.
Equation of a Cone
The equation \( x^2 + y^2 = z^2 \) represents a classical geometric surface known as a cone. A cone can be visualized in 3D as an infinite shape expanding outward from a central vertex, symmetrically spreading in all directions along the z-axis.
Features include:
  • Symmetry around the z-axis, with a circular cross-section in planes parallel to the base.
  • The equation derives from the Pythagorean theorem, where the radius at any level is equivalent to the height (z-value), forming right triangles vertically.
  • In 3D models, cones are important in rendering shadows and visual reflections accurately.
Recognizing the equation allows engineers and scientists to predict and utilize conical forms in structures, optical designs, and various simulations.

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Most popular questions from this chapter

Consider the "superparabolas" \(f_{n}(x)=x^{2 n}\) where \(n\) is a positive integer. a. Find the curvature function of \(f_{n^{\prime}}\) for \(n=1,2,\) and 3 b. Plot \(f_{n}\) and their curvature functions, for \(n=1,2,\) and \(3,\) and check for consistency. c. At what points does the maximum curvature occur, for \(n=1,2,\) and \(3 ?\) d. Let the maximum curvature for \(f_{n}\) occur at \(x=\pm z_{m}\) Using either analytical methods or a calculator, determine lim \(z_{n^{\prime}}\) Interpret your result.

Consider the following curves. a. Graph the curve. b. Compute the curvature. c. Graph the curvature as a ficnction of the parameter. d. Identify the points (if any) at which the curve has a maximum or minimum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. $$\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle, \text { for }-2 \leq t \leq 2 \quad \text { (parabola) }$$

Solving equations of motion Given an acceleration vector, initial velocity \(\left\langle u_{0}, v_{0}, w_{0}\right\rangle,\) and initial position \(\left\langle x_{0}, y_{0}, z_{0}\right\rangle,\) find the velocity and position vectors, for \(t \geq \mathbf{0}\). $$\mathbf{a}(t)=\langle 1, t, 4 t\rangle,\left\langle u_{0}, v_{0}, w_{0}\right\rangle=\langle 20,0,0\rangle, \left\langle x_{0}, y_{0}, z_{0}\right\rangle=\langle 0,0,0\rangle$$

The function \(f(x)=\sin n x,\) where \(n\) is a positive real number, has a local maximum at \(x=\frac{\pi}{2 n}\). Compute the curvature \(\kappa\) of \(f\) at this point. How does \(\kappa\) vary (if at all) as \(n\) varies?

Parabolic trajectories Show that the two-dimensional trajectory $$x(t)=u_{0} t+x_{0} \text { and } y(t)=-\frac{g t^{2}}{2}+v_{0} t+y_{0}, \text { for } 0 \leq t \leq T$$ of an object moving in a gravitational field is a segment of a parabola for some value of \(T>0 .\) Find \(T\) such that \(y(T)=0\).
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