Exam 14: Parametric Equations and Conic Sections

As $x \rightarrow-\infty$ , what does $\cosh x$ approach?

Let $x=e^{2 t}, y=e^{5 t},-\infty<t<\infty$ . a) Graph the curve. b) Eliminate the parameter to obtain an equation for $y$ as a function of $x$ .

Let an ellipse be parameterized by the equations x=-3+3t y=4+5t point a) What is the center? b) What is the length of the major axis? c) What is the length of the minor axis? d) Find an implicit equation of the curve.

Describe the graph of the parametric equations $x=2 \cos \left(\frac{\pi}{t+1}\right), y=2 \sin \left(\frac{\pi}{t+1}\right)$ for $t>$ 0.

Which of the following is the implicit equation for the ellipse graphed below?

A mouse hanging on to the end of the windmill blade shown below has coordinates given by $x=A \cos (k t), y=A \sin (k t)$ , where the origin is at the center of the blades, $x$ and $y$ are in meters, and $t$ is in seconds. The blades are 7 meters long and the windmill makes one complete revolution every 32 seconds in a counterclockwise direction. The mouse starts in the 3 o'clock position and, 4 seconds later, loses its hold and flies off. What is $k$ ? Round to 2 decimal places.

The equation of the parabola centered at the origin with focal point $(0,-2)$ is $y=\ldots x^{2}$ .

Could the graph below be the graph of the parametric equations $x=\cos t, y=2.2 \sin t$ ? Assume the parameter is restricted to values for which the functions are defined.

Give a parameterization for the straight line segment joining the points $(6,4)$ and $(-3,-2)$ .

Could the graph below be the graph of the parametric equations $x=e^{t}, y=t$ ? Assume the parameter is restricted to values for which the functions are defined.

Does $x=-1+2 \sin t, y=1-3 \cos t, 0 \leq t \leq 2 \pi$ , parameterize the ellipse graphed below, traversed in the clockwise direction and starting at the point $(-1,-2)$ ?

Write a parameterization for one branch of the following hyperbola using hyperbolic functions: $36 x^{2}+144 x=147,712+16 y^{2}+160 y, x>-2$

A bug starts at the point $(1,-1)$ and moves at 5 units\second along the $y$ -axis to the point $(1,2)$ . Then, the ant moves clockwise along a circle of radius 1 centered at $(1,1)$ to the point $(1,0)$ at a speed of 4 units\second. Express the bug's coordinates as a function of time, $t$ , in seconds.

Which of the following sets of equations parameterize the circle $x^{2}+y^{2}=25$ ? Mark all that apply.

Mercury moves in an elliptical orbit with the sun at a focus. The closest Mercury gets to the sun is 27 million miles and the furthest it gets from the sun is 41 million miles. The length of the major axis is ----------million miles.

Answer the following for the hyperbola $9 x^{2}+36 x=225-661+25 y^{2}+250 y$ : a) What is the center? b) Does the hyperbola open left-right or up-down? c) What are the values of $a$ and $b$ ?

Let $y=t$ be one part of a parametric equation. Which of the following formulas for $x$ as a function of $t$ on $0 \leq t \leq 6$ completes the parameterization of the graph of a straight line from $(0,0)$ to $(1,1)$ , and then continuing from $(1,1)$ in a straight line to $(0,6)$ ?

Write $\cosh (5 x)-\sinh (3 x)$ in terms of powers of $e$ .

An ellipse with horizontal and vertical axes is centered at the origin. If the point $(3,4)$ is on the ellipse, is the point $(-4,-3)$ necessarily on the ellipse as well?

Write a parameterization for one branch of the following hyperbola using hyperbolic functions: $\frac{y^{2}}{4}-(x-1)^{2}=1, y>0$