Unit I Exam: Review Guide
This page contains problems similar to those that will appear on the
Unit I Exam.
These questions are all similar to ones that you encountered while doing
your WebAssign homework.
1. Find the distance between the point \(P(1,1,1)\) and the line
\(\mathbf{r}(t) = \langle 3t, 1 - 2t, 2 + t \rangle\).
2. Find all three equations of the line through the points
\(P(0,\tfrac{1}{2}, 1)\) and \(Q(3,1,-4)\).
3. Determine whether the lines \(\ell_1\) and \(\ell_2\) are parallel,
skew, or intersecting. If intersecting, find the angle of intersection.
\[\begin{cases}
\mathbf{r}_1(t) = \langle 12+8t, 16-4t, 4 + 12t\rangle, \\
\mathbf{r}_2(s) = \langle 1+4s,3-2s,4+5s \rangle
\end{cases}\]
4. Find an equation of the plane through the points \(P(1,2,3)\),
\(Q(4,0,-1)\), and \(R(2,-4,-2)\).
5. Find the distance between the point and the given plane.
\[\begin{cases}
P(1,-3,2) \\
\Pi:\ \ 3x + 2y + 6z = 5
\end{cases}\]
6. Find an equation of the plane with \(x\)-intercept \(a\),
\(y\)-intercept \(b\), and \(z\)-intercept \(c\).
7. Reduce the equation to one of the standard forms and classify the
surface.
\[x^2 - y^2 - z^2 - 4x - 2z +3 = 0\]
8. You must be able to match the graph of a surface in \(\mathbb{R}^3\)
to its equation. (This will be a multiple choice question.)
9. Consider the vector function
\[\mathbf{r}(t) = \dfrac{t^2-1}{t-1}\,\mathbf{i} +
\sqrt{t+8}\,\mathbf{j} + \dfrac{\sin \pi t}{\ln t}\,\mathbf{k}.\]
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a.) |
What is the domain of \(\mathbf{r}\)?
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b.) |
Compute \(\displaystyle \lim_{t\to 1} \mathbf{r}(t)\), provided it exists.
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c.) |
Compute \(\dot{\mathbf{r}}(t)\), provided it exists.
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10. Find a parametrization of the curve of intersection of the cylinder
\(x^2 + y^2 = 4\) and the surface \(z = xy\).
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(Recall that "parametrization" is another name for a vector function whose
terminal points trace out the space curve.) |
11. Sketch the plane curve traced by the vector function
\[\mathbf{r}(t) = \left\langle e^t, e^{2t} \right\rangle.\]
12. Let \(\mathbf{u}\) and \(\mathbf{v}\) be vector functions in
\(\mathbb{R}^3\). Prove the product rule for the dot product:
\[ \frac{d}{dt}\Big[ \mathbf{u}(t)\cdot\mathbf{v}(t)\Big] =
\dot{\mathbf{u}}(t)\cdot\mathbf{v}(t) + \mathbf{u}(t)\cdot\dot{\mathbf{v}}(t).\]
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Hint. Write both \(\mathbf{u}\) and \(\mathbf{v}\) in coordinates. |
13. Consider the vector function
\(\mathbf{r}(t) = \left\langle \arctan(t), 2e^{2t}, 8te^t\right\rangle.\)
Find the unit tangent vector \(\mathbf{T}(0)\).
14. Find \(\mathbf{r}(t)\) if \(\dot{\mathbf{r}}(t) = \left\langle
t, e^t, te^t \right\rangle\) and \(\mathbf{r}(0) = \mathbf{i} + \mathbf{j} +
\mathbf{k}\).
15. Find symmetric equations for the line tangent to the space curve
\[\begin{cases}
x = e^{-t}\cos t, \\
y = e^{-t}\sin t, \\
z = e^{-t},
\end{cases}\]
at the point \(P(1,0,1)\).
16. If a space curve has the property that the position vector
\(\mathbf{r}\) is always perpendicular to the tangent vector \(\dot{\mathbf{r}}\),
show that the curve lies on a sphere centered at the origin. (Such space curves
are aptly call "spherical curves.")
17. Consider the space curve parametrized by
\[\mathbf{r}(t) = \left\langle 2t, t^2, \tfrac{1}{3}t^3 \right\rangle.\]
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a.) |
Find the arc length functional \(s = s(t)\) starting at the point \(t = 0\)
and moving in the positive \(t\) direction.
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b.) |
Use your answer from part a.) to compute the arc length of the curve
on the interval \(0\leq t \leq 1\).
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18. Reparametrize the plane curve
\[\mathbf{r}(t) = \left(\dfrac{2}{t^2-1} - 1\right)\,\mathbf{i} +
\dfrac{2t}{t^2+1}\,\mathbf{j}\]
with respect to arc length from the point \((1,0)\) in the direction of
increasing \(t\). Express the reparametrization in simplest form. What can you
deduce about the curve?
19. Compute the curvature \(\kappa\) of the twisted cubic
\(\mathbf{r}(t) = \left\langle t, t^2, t^3 \right\rangle\) at the point
\(P(1,1,1)\).
20.
Find the unit tangent, normal, and binormal vectors, and curvature of
the space curve
\[\mathbf{r}(t) = \left\langle t^2,\tfrac{2}{3}t^3,t \right\rangle\]
at the point \(P(1,\tfrac{2}{3},1)\).
21. At what point on the curve \(x = t^3\), \(y = 3t\), \(z = t^4\)
is the normal plane parallel to the plane \(6x + 6y - 8z = 1\)?
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Hint. The normal plane has normal vector \(\mathbf{T}\): it is spanned
by the normal and binormal vectors, \(\mathbf{N}\) and \(\mathbf{B}\). |
22. Show that the osculating plane at every point on the curve
\(\mathbf{r}(t) = \left\langle t+2, 1 - t, \tfrac{1}{2}t^2 \right\rangle\)
is the same plane. What can you conclude about the curve?
23.
Let \(C\) be a smooth space curve with unit tangent vector field \(\mathbf{T}\).
Prove that \(\mathbf{T} \perp \dot{\mathbf{T}}\) for all \(t\) in the domain
of \(\mathbf{T}\).
24.
Prove that the curvature of a circle of radius \(a\) is constant,
\(\kappa = 1/a \).
25.
Find an equation of the parabola (in the \(xy\)-plane) passing through the
origin and having osculating circle \( x^2 + (y + \tfrac{1}{2})^2 = \tfrac{1}{4}\)
at the origin.
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