Q. 1 - Calculus

Question

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

Step-by-Step Solution

Answer

True, True, True, True, True, True, True, True.

Part (a) Step 1. Given Information

The statement: 

Every curve in the plane has a unique

expression in terms of a vector-valued function with

two components.

Part (a) Step 2. Proving the statement to identify

With the help of an example: $r\left(t\right)=(\sin t)i+(\cos t)j$.

We can say that every curve in the plane has a unique

expression in terms of a vector-valued function with

two components, is true.

Part (b) Step 1. Given Information

The statement:
Every space curve has a unique expression in terms of a vector-valued function with three components.

Part (b) Step 2. Proving the statement to identify

With the help of an example: $r\left(t\right)=\ln t i +\sqrt{1-t} j+t k$
We can say that every space curve has a unique expression in terms of a vector-valued function with three components, is true.

Part (c) Step 1. Given Information

The statement:
The graph of two parametric equations

in two variables is also the graph of a vector-valued

function with two components.

Part (c) Step 2. Proving the statement to identify

With the help of an example:$x\left(t\right)=\cos t,y\left(t\right)=\sin t$ and $r\left(t\right)=(\sin t)i+(\cos t)j$
We can say that the graph of two parametric equations

in two variables is also the graph of a vector-valued

function with two components, is true.

Part (d) Step 1. Given Information

The statement:
Every vector-valued function in three variables can be expressed in terms of three parametric equations.

Part (d) Step 2. Proving the statement to identify

With the help of an example:$r\left(t\right)=(\sin t)i+(\cos t)j+tk$

Here, $x\left(t\right)=\cos t,y\left(t\right)=\sin t$ and $z\left(t\right)=t$

We can say that 

Every vector-valued function in three variables can be expressed in terms of three parametric equations, is true.

Part (e) Step 1. Given Information

The statement:
The graph of three parametric equations is also the graph of a vector-valued function with three components.

Part (e) Step 2. Proving the statement to identify

With the help of an example: $r\left(t\right)=(\sin t) i+(\cos t) j+tk$
We can say that 

The graph of three parametric equations is also the graph of a vector-valued function with three components, is true.

Part (f) Step 1. Given Information

The statement:
If $\mathit{r}\left(t\right)$ is a vector-valued function with domain $\mathrm{ℝ}$, $\underset{t\to c}{\lim }\mathit{r}\left(t\right)$ exists for every $c\in \mathrm{ℝ}$.

Part (f) Step 2. Proving the statement to identify

If $r\left(t\right)$ is a vector-valued function with domain $\mathrm{ℝ}$, $\underset{x\to c}{\lim }r\left(t\right)$ exists of every $c\in \mathrm{ℝ}$. 

Part (g) Step 1. Given Information

If a point $t_0$ is in the domain of a vector function $\mathit{r}\left(t\right)$, then $\underset{t\to t_0}{\lim }\mathit{r}\left(t\right)=\mathit{r}\left(t_0\right)$.

Part (g) Step 2. Proving the statement to identify

If a point $t_0$ is in the domain of a vector function $\mathit{r}\left(t\right),$then $\underset{t\to t_0}{\lim }\mathit{r}\left(t\right)=\mathit{r}\left(t_0\right)$, is true.

Part (h) Step 1. Given Information

The statement:
Every vector function is continuous at every point in its domain.

Part (h) Step 2. Proving the statement to identify

Every vector function is continuous at every point in its domain. If a point $t_0$ in its domain. So, it is true.