Find $dy/dx$ and $d^2y/dx^2$, and find the slope and concavity (if possible) at the given value of the parameter.

$$\underline{Parametric\ Equation}$$

$$x=\sqrt t,\ y=3t-1$$

$$\underline{Point}$$

$$t=1$$

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately $0.967$. The length of the major axis of the orbit is approximately $35.88$ astronomical units. (An astronomical unit is about $93$ million miles.)

Use a graphing utility to graph the equation of the orbit.

Convert the polar equation to rectangular form.

$$r=4$$

Graphical Reasoning Consider the parabola $x^2=4 p y$

Locate the focus for each parabola in part (a).

Determine whether the statement is true or false. Justify your answer.

In the standard form of the equation of a hyperbola, the larger the ratio of $b$ to $a$ the larger the eccentricity of the hyperbola.

Find $dy/dx$ and $d^2y/dx^2$, and find the slope and concavity (if possible) at the given value of the parameter.

$$\underline{Parametric\ Equation}$$

$$x=2t,\ y=3t-1$$

$$\underline{Point}$$

$$t=3$$

Graphical Reasoning Consider the parabola $x^2=4py$.

Use a graphing utility to graph the parabola for $p=1,p=2,p=3,\ \text{and}, p=4$ and Describe the effect on the graph when increases.

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately $0.967$. The length of the major axis of the orbit is approximately $35.88$ astronomical units. (An astronomical unit is about $93$ million miles.)

Find an equation of the orbit. Place the center of the orbit at the origin, and place the major axis on the $x$-axis.

Sketch the curve represented by the parametric equations $x=\sqrt{t}$ and y=1-t (indicate the direction of the curve) by eliminating the parameter and adjusting the domain of the resulting rectangular equation.

Convert the rectangular equation to polar form. Assume $a>0$.

$$y^3=x^2$$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

$$4x^2-6x-2y+7=0$$

The path of a softball is given by the equation

$$y=-0.08 x^2+x+4.$$

The coordinates x and y are measured in feet, with x=0 corresponding to the position from which the ball was thrown.

Analytically find the maximum height of the softball.

The path of a softball is given by the equation

$$y=-0.08 x^2+x+4.$$

The coordinates x and y are measured in feet, with x=0 corresponding to the position from which the ball was thrown.

Move the cursor along the path to approximate the highest point. Approximate the range of the trajectory.

Find $dy/dx$.

$$x=2e^\theta,y=e^{-\theta/2}$$