M210 Spring 93  Final Exam  May 4, 1993

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1. (20) i) Find the equation of the plane passing through the three points (0, 1, 2), (1, −1, 2), (2, 1, 0).

Express your answer in the form ax + by + cz = d.

Ans:  x+  y+  z =

1. ii) Find the angle that the diagonal of a cube makes with one of its edges.

Ans:

M210 Spring 93  Final Exam  May 4, 1993

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2. (15) Show that the equation of the tangent plane to the ellipsoid

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

at the point (x_{0}, y_{0}, z_{0}) has equation

\frac{xx_{0}}{a^{2}} + \frac{yy_{0}}{b^{2}} + \frac{zz_{0}}{c^{2}} = 1

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3. (15) The circular helix r⃗(t) = cos(t)i + sin(t)j + tk intersects the curve s⃗(t) = e^{t}i + tj + t^{2}k at the point (1, 0, 0). Find the angle of intersection of the two curves.

Ans:

M210 Spring 93  Final Exam  May 4, 1993

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4. (20) i). Let z = \frac{x}{1 + cos(y)}. Find \frac{∂z}{∂x} and \frac{∂z}{∂y}.

4. ii) Let z = uve^{u} where u = x^{2} − y^{2} and v = 2xy. Find \frac{∂z}{∂y}.

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5. (20). Consider

∫_{0}^{2} ∫_{y^{2}}^{6−y} f(x, y) dx dy.

i) Sketch the region of integration.

ii) Change the order of integration.

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6. (20) Evaluate

∫∫∫_{B} \frac{1}{1 + x^{2} + y^{2} + z^{2}} dV

where B is the ball of radius 1 centered at (0, 0, 0):

B = {(x, y, z)|x^{2} + y^{2} + z^{2} ≤ 1}.

Ans:

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7. (15) Evaluate

∫_{0}^{1} ∫_{0}^{√\overline{1−x^{2}}} e^{x^{2}+y^{2}}dy dx

by converting to polar coordinates.

Ans:

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8. (20) Consider the portion of the surface z = xy which lies within the cylinder x^{2} + y^{2} = 1.

i) Write down (at least one) integral which gives the surface area.

ii) Evaluate the integral to find the surface area.

Ans:

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9. (20) Find the absolute (overall) maximum value of the function

f(x, y, z) = xy + z

subject to the constraint

x^{2} + y^{2} + z^{2} ≤ 1.

You must find the critical point(s) in x^{2} + y^{2} + z^{2} < 1 and the maximum value(s) on x^{2} + y^{2} + z^{2} = 1.

The overall max value occurs at where f(x, y.z) = Name (Print) SSN:

10. (20) Let D be the upper half of the semicircle x^{2} + y^{2} ≤ 1. Find ȳ, the y–coordinate of the centroid of D. You may assume that the area is \frac{π}{2} .

Ans: ȳ =

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11. (15) Write the integral

∫_{−3}^{3} ∫_{−\sqrt{9−x^{2}}}^{+\sqrt{9−x^{2}}} ∫_{0}^{√\overline{9−x^{2}−y^{2}}}(x^{2} + y^{2} + z^{2})xy dz dy dx

as an integral in spherical coordinates. You need not evaluate the integral!