Q 1. True/False 9.3 - 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: If π 2 < θ < π, then the point (r, θ) is located in the second quadrant when it is plotted in a polar coordinate system.

(b) True or False: The graph of r = sin 5θ is a five-petaled rose.

(c) True or False: The graph of r = cos 6θ is a six-petaled rose.

(d) True or False: If a graph in the polar plane is symmetrical with respect to the origin, then for every polar point (r, θ) on the graph, the polar point (−r, θ +2π)

is also on the graph.

(e) True or False: The graph of a polar function r = f (θ) is symmetrical with respect to the y-axis if, for every point (r, θ) on the graph, the point (r,−θ) is also on the graph.

(f) True or False: When k is a positive integer, the polar roses r = sin kθ and r = cos kθ are symmetrical with respect to both the x-axis and y-axis if and only if k is even.

(g) True or False: In the rectangular coordinate system the graph of the equation (x 2+y2) 2 = k(x 2−y2) is a lemniscate for every k > 0.

(h) True or False: In the rectangular coordinate system the only function y = f (x) that is symmetrical with respect to both the y-axis and the origin is y = 0.

Step-by-Step Solution

Answer

(a) F

(b) T

(c) F

(d) T

(e) F

(f) T

(g) T

(h) T

Part (a) Step 1: Explanation

Consider the angle $\theta$ with an interval $\frac{\pi }{2}<\theta <\pi$

The second quadrant angles are represented by the $\theta$ interval.

However, the polar coordinate will have an endless number of representations for any point on the polar plane, $(r,\theta )$

As a result, the supplied statement is false.

Therefore, the answer is False.

Part (b) Step 1: Explanation

Consider $r=\sin 5\theta$

For the polar equation $r=\sin n\theta$ has $n$ petals when $n$ is an odd integer.

Thus the equation $r=\sin 5\theta$ has $5$ petals.

Therefore, the answer is True.

Part (c) Step 1: Explanation

Consider $r=\cos 6\theta$

For the polar equation $r=\cos n\theta$ has $2n$ petals when $n$ is an even integer.

The equation $r=\cos 6\theta$ yields $12$ petals as a result.

Therefore, the answer is False.

Part (d) Step 1: Explanation

A point in the polar plane is said to be symmetric with respect to the origin if the polar point $(-r,\theta +2\pi )$ is also on the graph for any point $(r,\theta )$

As a result, the given assertion is true.

Therefore, the answer is True.

Part (e) Step 1: Explanation

A point in polar plane is said to be symmetric with respect to $y$ if for any point $(r,\theta )$ on the graph, the polar point $(-r,-\theta )$ is also on the graph.

Thus, for any point $(r,\theta )$ the polar point $(r,-\theta )$ on graph then it not symmetric with respect to $y$ axis. 

Therefore, the answer is False.

Part (f) Step 1: Explanation

If the polar roses $r=\sin n\theta$ and $r=\cos n\theta$ are symmetrical with respect to both the axes $x, y$ for any positive integer $k$ then $k$ is even.

Therefore, the answer is True.

Part (g) Step 1: Explanation

For every rectangular coordinate system, the graph of the equation

${\left(x^2+y^2\right)}^2=k\left(x^2-y^2\right)$ is a lemniscuses.

Therefore the answer is True.

Part (h) Step 1: Explanation

In a $y$ rectangular coordinate system, the function is symmetrical with respect to the axis, and the origin is $y=0$

As a result, the given assertion is true.

Therefore, the answer is True.