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.

(a) True or False: Each point in the plane has a unique representation in rectangular coordinates.

(b) True or False: Each point in the plane has a unique representation in polar coordinates.

(c) True or False: If $b\ne c$, then the graphs of the polar equations $r=b$ and $r=c$ are different.

(d) True or False: If $\alpha \ne \beta$, then the graphs of the polar equations $\theta =\alpha$ and $\theta =\beta$ are different.

(e) True or False: The graph of $r=\csc \theta$ for $-\frac{\pi }{2}<\theta <\frac{\pi }{2}$ is a horizontal line in a polar coordinate system.

(f) True or False: In a polar coordinate system, the coordinates $(r,\theta )$ and $(r,\theta +\pi )$ represent the same point if and only if $r=0$.

(g) True or False: When $A$ and $B$ are nonzero constants, the graph of $r=A\sin \theta +B\cos \theta$ is a circle in a polar coordinate system.

(h) True or False: Every function $r=f\left(\theta \right)$ in a polar coordinate system can be expressed in terms of rectangular coordinates $x$ and $y$.

Step-by-Step Solution

Answer

(a). The statement given is True.

(b).  The statement given is False.

(c).  The statement given is False .

(d).  The statement given is False. 

(e).  The statement given is True.

(f).  The statement given is True .

(g).  The statement given is True .

(h).  The statement given is True .

part(a) step 1: Given information

Each point in the plane has a unique representation in rectangular coordinates.

part(a) step 2: Simplification

Consider a point $(-2,3)$ it is represented in the second quadrant the same point cannot be shown in the other quadrant.

Thus the point in rectangular plane has unique representation.

Therefore, the answer is True.

part(b) step 1: Given information

Each point in a plane has a unique representation in polar plane.

part(b) step 2: Simplification

Consider the point $\left(2,\frac{\pi }{3}\right)$, it is represented in the first quadrant the same point can be shown in the other quadrant $\left(2,\frac{4\pi }{3}\right)$.

Thus a point in polar plane rectangular plane has infinitely many representations.

Therefore the answer is False.

part(c) step 1: Given information

 If $b\ne c$, then the graphs of the polar equations $r=b$ and $r=c$ are different.

part(c) step 2: Simplification

If $b\ne c$ then the graphs of polar equations $r=b, r=c$ are not different in polar plane .

So the given statement is not correct.

Therefore, the answer is False.

part(d) step 1: Given information

If $\alpha \ne \beta$, then the graphs of the polar equations $\theta =\alpha$ and $\theta =\beta$ are different.

part(d) step 2: Simplification

If $\alpha \ne \beta$ then the graphs of the equations $\theta =\alpha$ and $\theta =\beta$ or not different in the plane.

Therefore, the answer is False.

part(e) step 1: Given information

The graph of $r=\csc \theta$ for $\frac{\pi }{2}<\theta <\frac{\pi }{2}$ is a horizontal line in a polar coordinate system.

part(e) step 2: Simplification

The graph of $r=\csc \theta$ is a horizontal line within the limits $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$.

Therefore, the answer is True

part(f) step 1: Given information

In a polar coordinate system, the coordinates $(r,\theta )$ and $(r,\theta +\pi )$ represent the same point if and only if $r=0$.

part(f) step 2: Simplification

In the polar coordinate system $(r,\theta )$ and $(r,\theta +\pi )$ represent the same point if and only if and only if $r=0$.

Therefore, the answer is True.

part(g) step 1: Given information

When $A$ and $B$ are nonzero constants, the graph of $r=A\sin \theta +B\cos \theta$ is a circle in a polar coordinate system.

part(g) step 2: Simplification

If $A, B$ are non zero constants then the graph $r=A\sin \theta +B\cos \theta$ is a circle in polar coordinate system.

Therefore, the answer is True.

part(h) step 1: Given information

Every function $r=f\left(\theta \right)$ can be expressed in rectangular coordinates $x, y$.

part(h) step 2: Simplification

Example: $r=2\cos \theta$ then $x^2+y^2=2·\frac{x}{r}$

$x^2+y^2=\frac{2x}{\sqrt{x^2+y^2}}$

Therefore the answer is True.