Parametric Equations, Polar Coordinates and Conic Sections

In this problem you will prove the three parts of Theorem 9.5:

(a) Prove that the polar coordinates (r, θ + 2πk) represent the same point for every integer k.

(b) Prove that the point with polar coordinates (−r, θ + π) represents the same point as (r, θ) for any value of θ.

(c) Prove that the polar coordinates (0, θ) represent the pole for any value of θ.

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: Every function $y = f \left(x\right)$ can be written in terms of parametric equations.

(b) True or False: Given parametric equations $x = x\left(t\right)$ and $y = y\left(t\right)$ the parameter can be eliminated to obtain the form y = f (x).

(c) True or False: Every parametric curve passes the vertical line test.

(d) True or False: Every curve in the plane has a unique expression in terms of parametric equations.

(e) True or False: If the functions x = f (t) and y = g(t) are differentiable for every t ∈ R, then the parametric curve defined by x and y is differentiable for every value of t.

(f) True or False: A curve parametrized by x = x(t), y = y(t) has a horizontal tangent line at (x(t0), y(t0)) if y (t) = 0.

(g) True or False: A curve parametrized by x = x(t), y = y(t) has a horizontal tangent line at (x(t0), y(t0)) if x (t) = 0 and y (t) = 0.

(h) True or False: The cycloid curve associated with a circle of radius r is made up of a series of semicircles of radius r.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Parametric equations $$x = f(t), y = g(t)$$ on the interval $$[0, 1)$$ that trace the unit circle exactly once clockwise, starting at the point $$(1, 0)$$.

(b) Parametric equations $$x = f(t), y = g(t)$$ on the interval $$[0, 2π)$$ that trace the circle centered at $$(2, −3)$$ with radius 5 exactly once counterclockwise, starting at the point $$(7, −3)$$.

(c) Parametric equations $$x = f(t), y = g(t)$$ whose graph is not the graph of a function $$y = f(x)$$.

If $x=x\left(t\right)$and $y=y\left(t\right)$are differentiable functions of $t$ determining the direction of motion along the curve when

(a) $x^{\text{'}}\left(t\right)>0$and$y^{\text{'}}\left(f\right)>0$.

(b) $x^{\text{'}}\left(t\right)>0$and $y^{\text{'}}\left(t\right)<0$.

(c) $x^{\text{'}}\left(t\right)<0$and $y^{\text{'}}\left(t\right)>0$

(d) $x^{\text{'}}\left(t\right)<0$and $y^{\text{'}}\left(t\right)<0$

4. Complete the following definition: Parametric equations are

Use the results of Exercise 3 to analyze the direction of motion for the parametric curves given by the equations in Exercises 5-8.

Use the results of Exercise 3 to analyze the direction of motion for the parametric curves given by the equations in Exercises 5–8. 

$x=t^2,y=t^3,t\in \mathrm{ℝ}$

Use the results of Exercise $3$3 to analyze the direction of

motion for the parametric curves given by the equations in

Exercises $5-8$

$x=\sin t,y=\cos t,t\in \mathrm{ℝ}$

Use the results of Exercise 3 to analyze the direction of motion for the parametric curves given by the equations in Exercises 5–8. 

$x=e^t,y=\ln t,t>0$

Use the results of Exercise 3 to analyze the direction of motion for the parametric curves given by the equations in Exercises 5–8 

$x=t^3-t,y=t^3+t^{3}t\in \mathrm{ℝ}$

parametrizations are provided for portions of the same function. For each problem do the following:

(i) Eliminate the parameter to show that the curves are portions of the same function. 

(ii) Describe the portion of the graph that each parametrization describes.

(iii) Discuss the direction of motion along with the graph for each parametrization.

$$\begin{gathered} \left(a\right) x = t, y ={ t}^2 - 1, t \ge 0 \\ \left(b\right) x = -t, y ={ t }^2- 1, t \ge 0 \end{gathered}$$

parametrizations are provided for portions of the same function. For each problem do the following:

(i) Eliminate the parameter to show that the curves are portions of the same function. 

(ii) Describe the portion of the graph that each parametrization describes.

(iii) Discuss the direction of motion along with the graph for each parametrization.

$$\begin{gathered} \left(a\right) x ={ t}^2, y = t^3, t \ge 0 \\ \left(b\right) x ={ t}^2, y = t^3, t \le 0 \end{gathered}$$

parametrizations are provided for portions of the same function. For each problem do the following:

(i) Eliminate the parameter to show that the curves are portions of the same function. 

(ii) Describe the portion of the graph that each parametrization describes.

(iii) Discuss the direction of motion along with the graph for each parametrization.

$$\begin{gathered} \left(a\right) x = t, y = \sin t, t \ge 0 \\ \left(b\right) x = t - 1, y = \sin (t - 1), t \ge 1 \end{gathered}$$

Suppose a parametric curve is given by parametric equations $x=x\left(t\right), y=y\left(t\right)$for $t$in some interval L. How can we find the slope of the parametric curve at some point $\left(x\left(t_0\right),y\left(t_0\right)\right)$? What is the equation of the tangent line to the parametric curve at the point $t_0$?

Explain how we can find the locations at which a parametric curve determined by $x = x\left(t\right)$ and $y = y\left(t\right)$ has horizontal or vertical tangent lines.

Show that the parametrization $x=2t+1,y=4t^2-4$ for $t\in [-1,\infty )$ has the same graph as the one we plotted point by point in the reading.

Explain why the parametrization $x=\sin t+1,y={\sin }^{2}t-4$for $t\in (-\infty ,\infty )$repeatedly traces the same small portion of the graph of the function $y=x^2-2x-3$.

sketch the parametric curve by plotting points.

$x = t, y = t^2, t \in R$

sketch the parametric curve by plotting points. 

$x=3t+1,y=1,t\in [-2,5]$

sketch the parametric curve by plotting points. $x=3+1,y=2t,t\in \lfloor 0,s\rfloor$

Sketch the parametric curve by plotting points. 

$x=2t-1,y=3t+5,t\in \mathrm{ℝ}$

 In Exercises 16–23 sketch the parametric curve by plotting points $\mathrm{x}={\mathrm{t}}^3-\mathrm{t},\mathrm{y}={\mathrm{t}}^3+\mathrm{t},\mathrm{t}\in \mathrm{ℝ}$

Sketch the parametric curve by plotting points. 

 $x=t^3-t,y=t^3+t,t\in \mathrm{ℝ}$

Sketch the parametric curve by plotting points. 

$x=2{\sin }^{3}t,y=2{\cos }^{3}t,t\in [0,2\pi ]$

In Exercises 16–23 sketch the parametric curve by plotting points. 

23. $x = {\cos }^{5}t , y = {\sin }^{5}t , t \in [0, 2\pi ]$

In Exercises 24–34 sketch the parametric curve by eliminating the parameter 

24. $x = 2t - 1, y = 3t + 5, t \in \mathrm{ℝ}$

Sketch the parametric curve by eliminating the parameter 

$x=2t-1,y=3t^2+5,t\in \mathrm{ℝ}$

sketch the parametric curve by eliminating the parameter 

$x=t+2,y=e^t,t\in \mathrm{ℝ}$

sketch the parametric curve by eliminating the parameter 

$x=\tan t,y=\tan t,t\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

Sketch the parametric curve by eliminating the parameter $x=\cos 2t,y=-\sin 2t,t\in [0,2\pi ]$

In Exercises 24-34 sketch the parametric curve by eliminating the parameter.

$x=3\cos t,y=4\sin t,t\in [0,2\pi ]$

In Exercises 24-34 sketch the parametric curve by eliminating the parameter.

$x=\sin t,y=\cos 2t,t\in [0,2\pi ]$

sketch the parametric curve by eliminating the parameter 

$x=\cosh t,y=\sinh t,t\in \mathrm{ℝ}$

In Exercises 24-34 sketch the parametric curve by eliminating the parameter.

\(x=\cosh t, y=\sinh t, t \in \mathbb{R}\)

In Exercises 24-34 sketch the parametric curve by eliminating the parameter.

$x=\csc t,y=\cot t,t\in (0,\pi )$

sketch the parametric curve by eliminating the parameter. 

$x={\log }_{10}t,y=\ln t,t\in (0,\infty )$

Finish Example 1 (b) by showing that the graph of the parametric equations $x=\cos t,y=\sin t,t\in [\pi ,2\pi ]$ is the bottom half of the unit circle centered at the origin with a counterclockwise direction of motion.

The curve is a circle centered at the origin. It is traced once, clockwise, starting at the point $(0,1)$ with $t\in [0,2\pi ]$.

The curve is a circle centered at the origin. It is traced once, counterclockwise, starting at the point $(0,3)$ with $t\in [0,1]$.

 The curve is a circle centered at the point $(a, b)$. It is traced once, counterclockwise, starting at the point $(a+r, b)$ with $t\in [0,2\pi ]$

The curve is a circle centered at the origin. It is traced once, counterclockwise, and contains all points of the unit circle except for $(0,-1)$ with $t\in \mathrm{ℝ}$.

Complete the calculation in Example 7 by using the trigonometric identity ${\sin }^2\left(\frac{\theta }{2}\right)=\frac{1}{2}(1-\cos \theta )$ to show that ${\int }_0^{2\pi }\sqrt{1-\cos \theta }d\theta =4\sqrt{2}$.

Find an equation for the line tangent to the parametric curve at the given value of t 

$x=2 t-1, y=3 t+5, t=-1$

find an equation for the line tangent to the parametric curve at the given value ot t. 

$x=t+2, y=e^t, t=0$

In Exercises 41-44 find an equation for the line tangent to the parametric curve at the given value ot f.

$x={\cos }^{3}t,y={\sin }^{3}t,t=\frac{\pi }{4}$.

In Exercises 41-44 find an equation for the line tangent to the parametric curve at the given value t f.

$x={\cos }^{3}t,y={\sin }^{3}t,t=\frac{\pi }{4}$.

In Exercises 45-48 use Example 6 to find $\frac{d^{2}y}{d{x}^2}$ for the parametric curve at the given value of t. Note that these are the same parametric equations as in Exercises 41-44.

$x=2 t-1, y=3 t+5, t=-1$.

In Exercises 45-48 use Example 6 to find $\frac{d^{2}y}{d{x}^2}$ for the parametric curve at the given value of t. Note that these are the same parametric equations as in Exercises 41-44.

$x=t+2,y=e^t,t=0$.

In Exercises 45-48 use Example 6 to find $\frac{d^{2}y}{d{x}^2}$ for the parametric curve at the given value of t. Note that these are the same parametric equations as in Exercises 41-44.

$x=t^2,y=(2-f{)}^2,t=\frac{1}{2}$.

Use the result of Exercise 54 to find parametric equations for the line segments connecting the given pairs of points in the direction indicated. 

From $(6,7)$to $(1,-3)$

Use the result of Exercise 54 to find parametric equations for the line segments connecting the given pairs of points in the direction indicated.  

 From $(0, c)$to $(-6, c)$