Calculus I, II, and III: A Problem-Based Approach with Early Transcendentals:

Section Practice

1. Complete the square for the parabola, $y=4x^2-8x+7\text{.}$ Sketch the parabola, labeling the vertex, focus, and directrix. Compute the area of the region bounded by the parabola, the x-axis, the vertical line through the focus, and $\dsp x={3 \over 2}\text{.}$

2. For the ellipse $3x^2+2y^2-12x+4y+2=0\text{,}$ find

   1. its center, vertices, foci, and the extremities of the minor axis;

   2. an equation of the tangent to the ellipse at $(0,-1)\text{.}$

3. For the ellipse $\dsp {{x^2} \over 9}+{{y^2} \over 4}=1\text{,}$

   1. set up a definite integral to evaluate the length of the arc of the ellipse from $(3,0)$ to $(-3,0)\text{;}$

   2. find an approximation of the arc length in part 1.

4. For the hyperbola $9y^2-4x^2+16x+36y-16=0\text{,}$

   1. find its center, vertices, foci, and equations of asymptotes;

   2. draw a sketch of the hyperbola and its asymptotes.

5. For each of the following equations, identify whether the curve is a parabola, circle, ellipse, or hyperbola by removing the $xy$ term from the equation by rotation of the axes.

   1. $x^2+2xy+y^2-4x+4y=0$

   2. $x^2+2xy+y^2+x-y-1=0$

   3. $24xy-7y^2-1=0$

   4. $8x^2-12xy+17y^2+24x-68y-32=0$

1. Find $\dsp{{dy} \over {\; dx}}$ and $\dsp {{d^2y} \over {dx^2}}$ for each of the following parametric curves.

   1. $x=a \cos(t)\text{,}$ $y =b \sin(t)$

   2. $x= \invsin(t)\text{,}$ $y = \invtan(t)$

   3. $x=t^ee^t\text{,}$ $y = t \ln(t)$ Only compute dy/dx for this one.

2. Let ${\cal C}$ be the curve defined by $x= \sin(t)+t\text{,}$ $y=\cos(t)-t\text{,}$ $t$ is in $\dsp \Big[-{{\pi} \over 2},{{\pi} \over 2} \Big]\text{.}$ Find an equation of the tangent to ${\cal C}$ at the point $(0,1)\text{.}$

3. Suppose we have an object traveling in the plane with position at time $t$ of $x=6-t^2$ and $y=t^2+4\text{.}$ When is the object stopped? When is it moving left? Right? Rewrite the parametric equations as a function by eliminating time.

4. Find the length:

   1. $x=e^t \cos(t)$ and $y=e^t \sin(t)$ from $t=1$ to $t=4$

   2. $x=4 \cos^3(t)$ and $y=4 \sin^3(t)$ from $t=\pi/2$ to $t= \pi$

5. Find the points of intersection of the curves ${\cal C}_1$ with parametric equations $x=t+1\text{,}$ $y=2t^2$ and ${\cal C}_2$ with parametric equations $x=2t+1\text{,}$ $y=2t^2+7\text{.}$ (Actuarial Exam)

6. Let ${\cal C}$ be the curve defined by $x= 2t^2+t-1\text{,}$ $y=t^2-3t+1\text{,}$ $-2\lt t\lt 2\text{.}$ Find an equation of the tangent to ${\cal C}$ at the point $(0,5)\text{.}$ (Actuarial Exam)

1. For each of the following problems, find a polar equation of the graph having the given equation in the rectangular coordinate system.

   1. $4x-3y=7$

   2. $xy=1$

   3. $x^3+y^3=3xy$

2. For each of the following problems, find an equation in the rectangular coordinate system having the given polar equation.

   1. $\dsp r = {{2} \over {1- 3\cos(\theta)}}$

   2. $r= 3 \sin(\theta)$

   3. $r^2=6 \cos(2 \theta)$

   4. $r=5 \sec(\theta)$

3. For each of the following problems, find the points of intersection of the graphs of the given pairs of polar curves.

   1. $r=4 \cos(2 \theta)\text{;}$ $r=2$

   2. $r= 1 - \cos(\theta)\text{;}$ $r= \cos(\theta)$

4. For each of the following problems, find the area of the region enclosed by the graph of the given polar equation.

   1. $r=4+4 \cos(\theta)$

   2. $r=4- \sin(\theta)$

   3. $r=2 \sin(3 \theta)$

5. For each of the following problems, find the area of the intersection of the region enclosed by the graphs of the two given polar equations.

   1. $r=4-2 \cos(\theta)\text{;}$ $r=2$

   2. $r=2 \sin(2\theta)\text{;}$ $r= 2 \cos(\theta)$

6. For each of the following problems, find the area of the inner loop of the limacon.

   1. $r=1-2 \cos(\theta)$

   2. $r=3-4 \sin(\theta)$