Math 344: Calculus III |
|
Instructions.
Complete all problems on your own paper,
showing enough work. You may use any resources that you'd like to
complete this review guide. Keep in mind that the final exam will
be closed notes. |
1. Find the curvature of the curve with parametric equations
$$
x = \int_0^t \sin\left(\tfrac{1}{2}\pi\theta^2\right)\, d\theta,\ \ \
y = \int_0^t \cos\left(\tfrac{1}{2}\pi\theta^2\right)\, d\theta.
$$
2. Show that the curve with vector equation
${\bf r}(t) = \left\langle a_1t^2 + b_1t + c_1, a_2t^2 + b_2t + c_2, a_3t^2
+ b_3t + c_3 \right\rangle$
lies in a plane, and find and equation of the plane.
3. Find the tangential and normal components of the acceleration
vector of a particle with position function
${\bf r}(t) = t\,{\bf i} + 2t\,{\bf j} + t^2\,{\bf k}$;
then write the acceleration vector ${\bf a}(t)$ as a linear combination of ${\bf T}$
and ${\bf N}$.
4. The helix ${\bf r}_1(t) = \langle \cos t, \sin t, t \rangle$
intersects the curve ${\bf r}_2(t) = \langle 1 + t, t^2, t^3 \rangle$ at the point
$P(1,0,0)$. Find the angle of intersection of these curves.
5. Reparametrize the curve ${\bf r}(t) = e^t\,{\bf i} +
e^t\sin t\,{\bf j} + e^t\cos t\, {\bf k}$ with respect to arc length measured from
the point $P(1,0,1)$ in the direction of increasing $t$.
6. Suppose $f$ is a differentiable function of one variable. Show that
all tangent planes to the surface $z = x\,f\left(\tfrac{y}{x}\right)$ intersect in a
common point.
7. Find all critical points of the function
$f(x,y) = x^3 - 6xy + 8y^3$
and determine whether they are local max, min, or saddle points of the function.
8. Find the maximum and minimum values of $f(x,y) = \dfrac{1}{x} +
\dfrac{1}{y}$ subject to the constraint $\dfrac{1}{x^2} + \dfrac{1}{y^2} = 1$.
9. Compute the second directional derivative $D^2_{\bf u}\, f(-2,0)$
in the direction of the point $(2,-3)$, where $f(x,y) = x^2y - y^2$.
10. Use a \(\delta\)-\(\varepsilon\) argument to prove that the limit does exist.
\[ \lim_{(x,y) \to (0,0)}\ \frac{5xy^2}{x^2 + y^2} \]
11. Show that when Laplace's equation
$$
\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} +
\frac{\partial^2 u}{\partial z^2} = 0
$$
is written in spherical coordinates, it becomes
$$
\frac{\partial^2 u}{\partial \rho^2} + \frac{2}{\rho}\frac{\partial u}{\partial \rho}
+ \frac{\cot \varphi}{\rho^2}\frac{\partial u}{\partial \varphi} +
\frac{1}{\rho^2}\frac{\partial^2 u}{\partial \varphi^2} + \frac{1}{\rho^2\sin^2\varphi}
\frac{\partial^2 u}{\partial \theta^2} = 0.
$$
Hint: Very carefully apply some chain rules.
12. The plane $\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1$, $a > 0$,
$b > 0$ , $c > 0$, cuts the solid ellipsoid $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} +
\dfrac{z^2}{c^2} \leq 1$ into two pieces. Find the volume of the smaller piece.
13. Find the area of the part of the cone $z^2 = a^2(x^2+ y^2)$ bounded
between the planes $z = 1$ and $z= 2$.
14. Compute the surface integral \(\displaystyle \int\!\!\!\int_S (x^2z + y^2z)\, dS\), where
\(S\) is the part of the plane \(z = 4 + x + y\) that lies inside of the cylinder \(x^2 + y^2 = 4\).
15. Evaluate \(\displaystyle \int\!\!\!\int_S {\rm curl}\,\mathbf{F}\cdot d\mathbf{S}\)
where \(\mathbf{F} = \langle x^2yz, yz^2, z^3e^{xy}\rangle\) and \(S\) is the part of the
sphere \(x^2 + y^2 + z^2 = 5\) that lies above the plane \(z = 1\). Take \(S\) to be oriented
upward. [Hint: Use Stokes' Theorem.]
16. Evaluate the path integral \(\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}\), where
\(\mathbf{F} = \langle xy, yz, zx \rangle\) and \(C\) is the triangle with vertices \((1,0,0)\),
\((0,1,0)\), and \((0,0,1)\), oriented counter-clockwise when looking "from above." [Hint: Use
Stokes' Theorem.]
17. Find the positively oriented Jordan curve $C$ for which the value of
the path integral $\displaystyle \int_C (y^3 - y)\, dx - 2x^3\, dy$ is a maximum.
18. Use the Divergence Theorem to calculate the flux of \(\mathbf{F}\) across
the surface \(S\), where \(\mathbf{F} = \langle x^3, y^3, z^3 \rangle\) and \(S\) is the
surface bounded by the cylinder \(x^2 + y^2 = 1\) and the planes \(z = 0\) and \(z = 2\).
19. Show that \(\mathbf{F}\) is a conservative vector field and use this
fact to evaluate the path integral, \(\displaystyle \int_C \mathbf{F}\cdot d\mathbf{r}\).
\[\begin{cases}
\mathbf{F} = (4x^3y^2 - 2xy^3)\mathbf{i} + (2x^4y - 3x^2y^2 + 4y^3)\mathbf{j}, \\[2 ex]
C: \mathbf{r}(t) = (t + \sin(\pi t))\mathbf{i} + (2t + \cos(\pi t))\mathbf{j},\ \ \ 0 \leq t \leq 1.
\end{cases}\]
20. Suppose \(f\) is a harmonic function on an open domain \(D \subseteq \mathbb{R}^2\);
that is, \(\Delta f = 0\) on \(D\). Show that \(\displaystyle \int_C \partial_y f\, dx -
\partial_x f\, dy\) is independent of path in \(D\).
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