Textbook Problem

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Investigation Consider the function in Exercise 14.

(a) Sketch a graph of the function where u is held constant at $u=1$. Identify the graph.

(b) Sketch a graph of the function where v is held constant at $v=2\pi /3$. Identify the graph.

(c) Assume that a surface is represented by the vector-valued function $\text{r}=\text{r(}u,v\text{)}$. What generalization can you make about the graph of the function when one of the parameters is held constant?

To determine

To graph: The function, $r\left(u,v\right)=2u\cos vi+2u\sin vj+vk$, $0\le u\le 1,0\le v\le 3\pi$ where $u$ held constant at $u=1$ and identify the graph.

Explanation of Solution

Given:

The vector-valued function,

$r\left(u,v\right)=2u\cos vi+2u\sin vj+vk$, $0\le u\le 1,0\le v\le 3\pi$

The value of $u$ is constant and that is $1$.

Graph:

The provided function,

$r\left(u,v\right)=2u\cos vi+2u\sin vj+vk$

If, $u=1$ then the function becomes,

$r\left(u,v\right)=2\cos vi+2\sin vj+vk$

Now, compare the above function with general from of parametric equation surface,

$r\left(u,v\right)=x(u,v$

To determine

To graph: The function, $r\left(u,v\right)=2u\cos vi+2u\sin vj+vk$, $0\le u\le 1,0\le v\le 3\pi$ where $v$ held constant at $v=\frac{2\pi }{3}$ and identify the graph.

To determine

The generalization that can made about the graph of the function $r\left(u,v\right)=2u\cos vi+2u\sin vj+vk$, $0\le u\le 1,0\le v\le 3\pi$ when one of the parameters is held constant.