Section Practice
¶These are practice problems for the tests. Solutions are in the next section. While we will not present these, I am happy to answer questions about them in class.
Conics
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 \(\dsp x={3 \over 2}\text{.}\)
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For the ellipse \(3x^2+2y^2-12x+4y+2=0\text{,}\) find
its center, vertices, foci, and the extremities of the minor axis;
an equation of the tangent to the ellipse at \((0,-1)\text{.}\)
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For the ellipse \(\dsp {{x^2} \over 9}+{{y^2} \over 4}=1\text{,}\)
set up a definite integral to evaluate the length of the arc of the ellipse from \((3,0)\) to \((-3,0)\text{;}\)
find an approximation of the arc length in part 1.
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For the hyperbola \(9y^2-4x^2+16x+36y-16=0\text{,}\)
find its center, vertices, foci, and equations of asymptotes;
draw a sketch of the hyperbola and its asymptotes.
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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.
\(x^2+2xy+y^2-4x+4y=0\)
\(x^2+2xy+y^2+x-y-1=0\)
\(24xy-7y^2-1=0\)
\(8x^2-12xy+17y^2+24x-68y-32=0\)
Parametrics
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Find \(\dsp{{dy} \over {\; dx}}\) and \(\dsp {{d^2y} \over {dx^2}}\) for each of the following parametric curves.
\(x=a \cos(t)\text{,}\) \(y =b \sin(t)\)
\(x= \invsin(t)\text{,}\) \(y = \invtan(t)\)
\(x=t^ee^t\text{,}\) \(y = t \ln(t)\) Only compute dy/dx for this one.
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 \((0,1)\text{.}\)
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.
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Find the length:
\(x=e^t \cos(t)\) and \(y=e^t \sin(t)\) from \(t=1\) to \(t=4\)
\(x=4 \cos^3(t)\) and \(y=4 \sin^3(t)\) from \(t=\pi/2\) to \(t= \pi\)
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)
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)
Polars
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For each of the following problems, find a polar equation of the graph having the given equation in the rectangular coordinate system.
\(4x-3y=7\)
\(xy=1\)
\(x^3+y^3=3xy\)
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For each of the following problems, find an equation in the rectangular coordinate system having the given polar equation.
\(\dsp r = {{2} \over {1- 3\cos(\theta)}}\)
\(r= 3 \sin(\theta)\)
\(r^2=6 \cos(2 \theta)\)
\(r=5 \sec(\theta)\)
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For each of the following problems, find the points of intersection of the graphs of the given pairs of polar curves.
\(r=4 \cos(2 \theta)\text{;}\) \(r=2\)
\(r= 1 - \cos(\theta)\text{;}\) \(r= \cos(\theta)\)
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For each of the following problems, find the area of the region enclosed by the graph of the given polar equation.
\(r=4+4 \cos(\theta)\)
\(r=4- \sin(\theta)\)
\(r=2 \sin(3 \theta)\)
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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.
\(r=4-2 \cos(\theta)\text{;}\) \(r=2\)
\(r=2 \sin(2\theta)\text{;}\) \(r= 2 \cos(\theta)\)
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For each of the following problems, find the area of the inner loop of the limacon.
\(r=1-2 \cos(\theta)\)
\(r=3-4 \sin(\theta)\)