Review for Exam I
Example 1. (a) Find the equation of the plane which contains the point (2, 1, −1) and
is perpendicular to the planes 3x + y − z = 6 and 2x + 2y + z = 4.
(b) Find the distance from the point (1, 2, 3) to the plane solved in (a).
Example 2. Given the surface z 2 + x2 − y 2 − 6z + 2x + 12 = 0. Reduce the equation
to the standard form of quadric surfaces and classify the surface. You don't need to
sketch the graph, just identify the type of surface.
Example 3. Let C be the curve with vector equation r(t) = h 4t, t2 , 4 ln t i.
(a) Find equations for the tangent line to C at the point (4, 1, 0).
(b) Find the arc length of the curve from t = 1 to t = 2.
Example 4. (a) Find and sketch the domain of f (x, y) =
√
xy − 1.
(b) Find an equation of the tangent plane to the graph of this function at the point
(x, y) = (2, 2).
Example 5. Use dierentials to approximate the number
√
3.022 + 3.972 .
∂ 2z
.
Example 6. Let z = x sin y , where x = ut and y = u + t. Find
∂u∂t
Example 7. Find zx and zy if the function z = z(x, y) is given implicitly by the
following equation
x2 zexy = cos(xy + z)