MA261 Fall 2018 quiz

Which of the following pairs of planes are orthogonal to each other?

• x + 10y − z = 6, −9x − y − 19z = 2

• 5x + 8y = −3, y + 6z = 1

• x = 5z + 3y, 8x − 6y + 2z = −1

• 8x + 5y = −3, 9y + 6z = −1

• 8x + 5y = −3, y + 6z = −1

Which of the following equations produces a surface that is NOT shown here?

• $-x^2+y^2-z^2=1$

• $9x^2+4y^2+z^2=1$

• $y=x^2-z^2$

• $x^2-y^2+z^2=1$

• $y=2x^2+z^2$

Find a so that the point (3, a, 1) is on the tangent plane to  $z=e^{xy}-4x^2y+3y^2$  at (0,1,4).

•  $\frac{1}{2}$  

•  $-\frac{1}{2}$  

•  $-\frac{1}{7}$  

• 0

•  $\frac{1}{6}$  

Find the directional derivative of  $f\left(x,y\right)=\sqrt{4x^2+3y}$  at (2,3) in the direction of  $\overrightarrow{i}-2\overrightarrow{j}$  .

•  $\frac{1}{5}$  

•  $\frac{2}{5}$  

•  $\frac{1}{\sqrt{5}}$  

•  $\frac{11}{\sqrt{5}}$  

•  $\frac{11}{5}$  

For the level surface $3y^2z+xz^2=10$  find  $2\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$  at (1,-1,2)

•  $\frac{4}{5}$  

•  $\frac{20}{7}$  

•  $\frac{4}{7}$  

•  $\frac{1}{5}$  

•  $-\frac{4}{7}$  

Find the minimum value of f(x, y) = 2x + 3y + 2 given that $2x^2+5xy+4y^2=28$  

• -1

• -2

• -3

• -6

• -8

Let $f\left(x,y\right)=\left(x^2+y^2\right)e^x$  . This function has

• a local max. and a local min. point

•  two local max. points 

• a local max. and a saddle point

•  two local max. points 

• a local min. and a saddle point

Let D be the region in the first quadrant between the circles $x^2+y^2=1$  and  $x^2+y^2=4$ . Evaluate the integral $\int\int_D\frac{\left(x^2y\right)}{\left(x^2+y^2\right)^{\frac{3}{2}}}dA$  

•  $\frac{10}{3}$  

•  $\frac{1}{2}$  

•  $\frac{3}{2}$  

•  $\frac{14}{3}$  

•  $\frac{5}{6}$  

Which of the following integrals represents the volume of the solid in the first octant that is bounded on the side by the surface $x^2+y^2=4$  and on the top by the surface  $x^2+y^2+z^2=4$  ?

•  $\int_0^2\int_0^2\int_0^{\left(4-x^2-y^2\right)}dzdxdy$  

•  $\int_0^4\int_0^{\sqrt{4-x^2}}\int_0^{\left(4-z\right)}dzdydx$  

•  $\int_0^2\int_0^{\sqrt{4-x^2}}\int_0^{\left(4-x^2-y^2\right)}dzdydx$  

•  $\int_0^2\int_0^{\sqrt{4-x^2}}\int_0^{\left(x^2+y^2\right)}dzdydx$  

•  $\int_0^4\int_0^4\int_0^{\left(4-x^2-y^2\right)}dzdxdy$  

Convert the integral to cylindrical, then evaluate it: $\int_{-2}^2\int_0^{\sqrt{4-x^2}}\int_0^{\left(4-x^2-y^2\right)}15\sqrt{x^2+y^2}dzdydx$  

• 4π

• 16π

• 32π

• 43π

• 64π