Parametric Equations and Polar Coordinates: True-False Quiz and Solutions (Notes)

in mathematics •  6 months ago

In this video I go over an extensive recap on Polar Equations and Polar Coordinates by going over the True-False Quiz found in the end of my Stewart Calculus textbook. This is a 10-Part quiz which involves evaluating each statement to determine if it is true or false, and explaining why and/or providing an example that makes it false. As many of you know by now, I have been busy researching into Suppressed Science, or what I like to call Real Science, so this video serves as a good way to take a break from my research and to get back into the groove of my mathematically challenging tutorials. This video is much more extensive than my usual math tutorials and this is because I plan to spend more time researching and less time producing individual videos. Nonetheless, I will look to cover MUCH MORE math concepts and topics than in previous years by combining them into less videos, but of longer duration.

The True-False Quiz I cover involves graphing, differentiating, integrating, and comparing functions in Cartesian, Polar, and Parametric form; including Conic Sections in Polar Coordinates. This is a very extensive and very great way to get caught up on Parametric and Polar Equations in general so make sure to watch this video!

MES NOTE: In Question/Statement 10 of the Quiz I also go over a mistake from my earlier video regarding the Hyperbola and its location of the Directrix. The Directrix should be in between the Focus and Center; and NOT as shown in my earlier video: https://steemit.com/mathematics/@mes/conics-in-polar-coordinates-unified-theorem-hyperbola-proof

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Parametric Equations and Polar Coordinates: True-False Quiz and Solutions

At the end of each chapter of my Calculus Textbook there is a "True-False Quiz" which provides a good recap to test your understanding of the topics covered in that chapter. And since I have been stuck in the wonderful world of #FreeEnergy, #AntiGravity, and #Gyroscopes for these past few months, going over this quiz will be a good recap to get back into the groove of my Mathematics Video Tutorials… and a break from my #SuppressedScience videos. Anyways, let's jump right in!

True-False Quiz

Determine whether the statement is true or false.

If it is true, explain why.

If it is false, explain why or give an example that disproves the statement.

Q1. If the parametric curve x = f(t), y = g(t), satisfies g'(t) = 0, then it has a horizontal tangent when t = 1.

Q2. If x = f(t) and y = g(t) are twice differentiable, then d2y / dx2 = (d2y / dt2) / (d2x / dt2).

Q3. The length of the curve x = f(t), y = g(t), a ≤ t ≤ b, is:

Q4. If a point is represented by (x , y) in Cartesian coordinates (where x ≠ 0) and (r , θ) in polar coordinates, then θ = tan-1(y/x).

Q5. The polar curve r = 1 - sin2θ and r = sin2θ - 1 have the same graph.

Q6. The equation r = 2, x2 + y2 = 4, and x = 2 sin3t, y = 2 cos3t (0 ≤ t ≤ 2π) all have the same graph.

Q7. The parametric equations x = t2 , y = t4 have the same graph as x = t2 , y = t6.

Q8. The graph of y2 = 2y + 3x is a parabola.

Q9. A tangent line to a parabola intersects the parabola only once.

Q10. A hyperbola never intersects its directrix.

Solution:

Q1: If the parametric curve x = f(t), y = g(t), satisfies g'(t) = 0, then it has a horizontal tangent when t = 1.

Recall that a typical graph of a parametric curve is shown below:

From the dy/dx formulation above it is clear that if g'(t = 1) = 0, then the parametric curve has a horizontal tangent ONLY if f'(t) is NOT equal to 0; in which case it would not be defined. Thus Statement 1 is FALSE!

For example consider the parametric curve:

https://www.desmos.com/calculator/ah5wgv3rtv

Retrieved: 4 July 2018
Archive: Not Available

Note the sharp point at the origin where the Vertical Tangent occurs.

Note: If the Statement stated instead that f'(1) ≠ 0 when g'(t) = 0, then it would be considered TRUE.

Q3. The length of the curve x = f(t), y = g(t), a ≤ t ≤ b, is:

Recall that this is the length of a curve from t = a to t = b, but with parametric curves the SAME path can be traced multiple times. Thus the figure obtained from the above equation does not necessarily equal the length of the curve, but the path a "particle" takes on that curve. Thus this Statement is FALSE!

Consider the following example:

Since t increases from 0 to 4π, the circle is traversed twice and the integral gives TWICE the length (or twice the circumference of the circle).

https://www.desmos.com/calculator/wgmhvzcypk

Retrieved: 6 July 2018
Archive: Not Available

Notice how the particle traces the circle twice! #Amazing

Q5. The polar curve r = 1 - sin2θ and r = sin2θ - 1 have the same graph.

Thus in Polar Coordinates, the curves r = 1 - sin2θ and r = sin2θ - 1 are the same curves. Thus the Statement is TRUE!

https://www.desmos.com/calculator/bkp4mbhukj

Retrieved: 3 July 2018
Archive: Not Available

Note that in Cartesian coordinates the two curves are not the same. #VeryInteresting

Also note that in polar coordinates r represents the distance from the pole, regardless if the function gives positive or negative values!

Q6. The equations r = 2, x2 + y2 = 4, and x = 2 sin3t, y = 2 cos3t (0 ≤ t ≤ 2π) all have the same graph.

Thus all the equations are circles of radius r = 2; thus the Statement is TRUE!

MES Note: This is assuming that the parametric circle looping around 3 times counts as "the same graph"; since the polar equation r = 2 can technically represent a circle with INFINITE loops too.

https://www.desmos.com/calculator/qz9yw1w62r

Retrieved: 3 July 2018
Archive: Not Available

Notice how all the graphs pile up on top of each other since they represent the same circle!

Note that the parametric equations form a complete circle at a minimum interval of 0 ≤ t ≤ 2π/3. Lowering this interval to 0 ≤ t ≤ 2π/4 obtains just 3 quarters of a circle. #Interesting!

Q8. The graph of y2 = 2y + 3x is a parabola.

Recall that a shifted horizontal parabola is of the form:

Let's simplify first and complete the square:

https://www.desmos.com/calculator/p1bqqfuhub

Retrieved: 3 July 2018
Archive: Not Available

The Statement is TRUE as can be seen from this horizontal right opening parabola!

Q9. A tangent line to a parabola intersects the parabola only once.

MES Note: This question actually is meant to say a tangent line "meets" a parabola once, and not necessarily moving "through" it.

We can actually see this visually:

Note that the parabola gets further away from the tangent line as you move further from the point of intersection. Thus the Statement is TRUE!

We can also see this numerically. Assume we have a parabola of the form:

Let's find a line tangent to the parabola at the point x = a.

We can determine the points this Tangent line meets the parabola by setting the two equations equal to each other and solving for x.

This shows that each tangent line meets the parabola at exactly one point. Thus the Statement is TRUE once again!

Q10. A hyperbola never intersects its directrix.

Recall the Unified Theorem for Conics Sections determines a Hyperbola for the case where the "eccentricity" e > 1:

Thus the point P is closer to the Directrix L than it is to the Focus F at all points on the Hyperbola.

Note that the opposite side of the hyperbola ALSO won't intersect the Directrix; because if the hyperbola ever does intersect it then it won't be considered a hyperbola in the first place!

MES Note: As will be explained further in this video, note that for a positive directrix (x>0), then it will be located to the LEFT of the Hyperbola center!

Thus since e would vary from e > 1 and e < 1 if the curve intersects the Directrix then the curve is NOT a hyperbola!

Thus the Statement is TRUE and a hyperbola never intersects its directrix!

We can also see this is graphically the case by considering the Polar Equation of the Unified Theorem for Conic Sections:

https://www.desmos.com/calculator/eau7redi9d

Retrieved: 3 July 2018
Archive: Not Available

Notice how the Directrix line never makes contact with the Hyperbola.

When d = x = 0, the equation becomes r = 0, which is thus just a point at the origin; i.e. NOT a hyperbola.

Note also that varying e (written as a in the Desmos Calculator since e is already defined as the famous number 2.71828…) also shows the directrix closer to the focus than the center of the hyperbola.

Fascinating stuff!

MES Note: In my earlier video on the Hyperbola Proof for the Unified Conic Sections Theorem, I had made the following misleading drawing:

https://steemit.com/mathematics/@mes/conics-in-polar-coordinates-unified-theorem-hyperbola-proof

Retrieved: 4 July 2018
Archive: https://archive.li/GL6KT

This is actually misleading and the Directrix should be moved to the LEFT of the Hyperbola Center point and not intersecting the Hyperbola!

From my earlier derivation also recall that:

Thus, the following figure correctly describes where the Directrix is located, assuming the Focus used in the Polar Equation is at the origin.

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I usually just scroll through and press the vote buton, but this was a good read. Thanks for not wasting my time!