Math 344: Calculus III |
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Instructions.
Complete all problems on this paper,
showing enough work. You may use any resources that you'd like to
complete this review guide. Keep in mind that the midterm exam will
be closed notes. |
1-4. True/False [1 point each] Determine whether each statement
is either always true, or false. If you determine that the
statement is false, you must give justification to receive credit.
1. Let ${\bf r}$ be a smooth vector function. If $\Vert {\bf r}(t)
\Vert = 1$ for all $t$, then $\Vert \dot{\bf r}(t)\Vert$ is constant.
2. Let ${\bf r}$ be a smooth vector function. If $\Vert {\bf r}(t) \Vert = 1$
for all $t$, then $\dot{\bf r}(t)$ is orthgonal to ${\bf r}(t)$ for all
$t$.
3. Let ${\bf r}$ be a vector function such that $\ddot{\bf r}(t)$ exists for
all $t$, $\dot{\bf r} \neq {\bf 0}$, and $\ddot{\bf r} \neq {\bf 0}$. Then
${\bf N} = \kappa\, {\bf T}$.
4. Let $f$ be a function of $(x,y)$. If $f$ has a local minimum at $(a,b)$,
then $\nabla f(a,b) = {\bf 0}$.
5. [1 point]
Find a vector function that represents the curve of intersection of the
cylinder $x^2 + y^2 = 16$ and the paraboloid $z = 2x^2 + 2y^2$.
6. [1 point]
Find the point on the curve $y = e^{2x}$ at which the curvature is
maximized.
7. [3 points]
Find equations of the osculating circles to the curve $y = x^4 - x^2$ at
each critical point.
8. [1 point]
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 arclength from the point $(1,0,1)$ in
the direction of increasing $t$.
9. [1 point]
Use your answer to problem 8 to compute the curvature of the space
curve as a function of arclength.
10. [3 points]
Find the unit tangent, normal, and binormal vectors to the curve
${\bf r}(t) = \langle \sin^3 t, \cos^3 t, \sin^2 t \rangle$ at the point
$t = \pi/4$.
11. [1 point]
Consider the function $f(x,y) = x^2 - 2x + y^2 + y$, and let $C$ denote
the level curve $f(x,y) = 5$. Show that the gradient vector $\nabla f(-1,1)$
is orthogonal to $C$ at the point $(-1,1)$.
12. [1 point]
Let $z = \sin\big(x + \sin t\big)$. Show that $\dfrac{\partial z}{\partial x}\,
\dfrac{\partial^2 z}{\partial x \partial t} = \dfrac{\partial z}{\partial t}\,\dfrac{\partial^2 z}{\partial x^2}$.
13. [2 points]
Find the absolute maximum and minimum values of the function
$f(x,y) = e^{-x^2-y^2}(x^2 + 2y^2)$ on the domain $D:
\big\{(x,y) \mid x^2 + y^2 \leq 4\big\}$.
14. [1 point]
Use differentials or a linear approximation to estimate the number
$(1.98)^3\sqrt{(3.01)^2 + (3.97)^2}$.
15. [1 point]
Let $f(x,y) = x^2e^{-y}$. Compute the second directional derivative
$D_{\bf v}^2\,f(-2,0)$, where ${\bf v} = \langle -3,4 \rangle$.
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