ENGR 251 Final: ENGR 251 Final Exam 2006 Winter Solutions
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1 = cos(zx)ez+y (a) find the equation of the tangent plane at the point ( , 0, 0) Solution: let f (x, y, z) = cos(zx)ez+y. Then the surface given by 1 = cos(zx)ez+y is the level surface {(x, y, z) : f (x, y, z) = 1}. To nd the tangent plane at a point we rst need to nd a normal vector to the surface at that point, which is given by the gradient. Computing the partial derivatives (using the chain rule, the product rule and the fact that ez+y = ezey), we get: Plugging in x = , y = 0, z = 0 gives: ~ f ( , 0, 0) = 0 sin(0)e0i + cos(0)e0j + ( sin(0)e0 + cos(0)e0)k = 0i + 1j + ( 0 + 1)k = j + k. This is a normal vector to the plane.
