Midterm Review

Sample Problems

Here are some problems that may help you prepare for the midterm.

1) Find the center of the sphere with equation $x^2+20x+y^2-20y+z^2-46z=0$.

2) Find the equation of the plane through the points $(0,0,-4)$, $(0,-7,0)$, and $(-5,0,0)$.

3) True or False: If $z=\displaystyle 4e^{-\left(9x^2+9y^2\right)}$, there is a level curve for every value of $z$.

4) Represent the paraboloid obtained by shifting $z=x^2+y^2$ vertically 5 units: (1) as a graph of the function $f(x,y)$ and, (2) as a level surface of the form $g(x,y,z)=c$. (There are many possible answers.)

5) True or False: The level surface $8x+6y+7z-14=0$ can be expressed as the graph of a function $f(x,y)$.

6) True or False: The level surface $3x^2+3y^2+3z^2-3=0$ can be expressed as the graph of a function $f(x,y)$.

7) Represent the plane with intercepts $x=2$, $y=3$, and $z=4$: (1) as a graph of the function $f(x,y)$ and, (2) as a level surface of the form $g(x,y,z)=c$. (There are many possible answers.)

8) Do exercises 5-13 and problems 24 and 30 in section 12.3.

9) Find the limit $\displaystyle\lim_{(x,y)\to(0,0)}\frac{2x+4y}{(\sin y)-8}$ or prove that it does not exist.

10) Are the planes $4x+6y-2z=4$ and $f(x,y)=2x+3y$ parallel? Are they perpendicular?

11) Find an equation of the plane parallel to $z=1-x+6y$ that contains the point $(1,1,1)$.

12) True or False: The value of $\vec{v}\cdot(\vec{v}\times\vec{w})$ is always zero.

13) True or False: The triangle in 3-space given by the vertices $(4,1,0)$, $(0,1,0)$, and $(0,1,1)$ contains a right angle.

14) True or False: $\vec{v}\times\vec{w}$ is NEVER the same as $\vec{v}\cdot\vec{w}$.

15) Given $\vec{v}=3\vec{i}+3\vec{j}+4\vec{k}$, find the value of $t$ such that the vector $t\vec{i}+2t\vec{j}+3\vec{k}$ is perpendicular to $\vec{v}$.

16) If $z=\displaystyle\frac{2x^2y^6+5y^6}{15xy-5}$, find $z_y$.

17) Find $\displaystyle\frac{\partial}{\partial x}\left(xe^{\sqrt{6xy}}\right)$

18) Find $f_x(x,y)$ and $f_y(x,y)$ for $f(x,y)=\arctan\left(\frac{y}{x}\right)$ and $f(x,y)=y\log_x(y)$.

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