Question How many variables are used to parameterize a surface in R^3? What kind of geometric object is the boundary of a surface? How many variables are used to parameterize the boundary of a surface? Find a parametric representation for the upper half, Z ≥ 0, of the surface 2x^2 + 3y^2 + z^2 = 6.

(1 point) For the curve given by r(t) = (6t, -3t, 1 – 7ta), Find the derivative r' (t) = / > Find the second derivative p" (t) = ( 2 > Find the curvature at t=1 K(1) =

Minimum sum Find positive numbers x and y satisfying the equation xy = 12 such that the sum 2x + y is as small as possible.

The SP Corporation makes 41,000 motors to be used in the production of its sewing machines. The average cost per motor at this level of activity is: Direct materials Direct labor $ 10.00 $9.00 $ 3.70 $ 4.65 Variable manufacturing overhead Fixed manufacturing overhead An outside supplier recently began producing a comparable motor that could be used in the The SP Corporation makes 41,000 motors to be used in the production of its sewing machines. The average cost per motor at this level of activity is: Direct materials Direct labor $ 10.00 $9.00 $ 3.70 $ 4.65 Variable manufacturing overhead Fixed manufacturing overhead An outside supplier recently began producing a comparable motor that could be used in the sewing machine. The price offered to SP Corporation for this motor is $25.45. If SP Corporation decides not to make the motors, there would be no other use for the production facilities and none of the fixed manufacturing overhead cost could be avoided. Direct labor is a variable cost in this company. The annual financial advantage (disadvantage) for the company as a result of making the motors rather than buying them from the outside supplier would be: (2.5%) 1) $112,750 2) $264,450 3) $190,650 4) ($77,900)

Find the equation of the plane that passes through the point (2, 1,2) and is tangent to the surface x² + 4y + 3z? = 20 at (2, 1,2). Write it in the form indicated below. Equation: 4(x – 2)+ (y- (z- )= 0

Find the equation of the plane tangent to apy2 + 2-9= 0 at x = 1, y = 2. The equation of the plane is

(1 point) Show that S = {0,4, 8, 12, 16, 20} is a subring of Z24 by showing that it satisfies the requirements of the subring test: (i) nonempty (clearly satisfied), (ii) closed under subtraction (completely the first table below), (iii) closed under multiplication (complete the second table below). (ii) Show that S is closed under subtraction by completing the following table, where the body of the table contains "a - b", where a is the element in the left column, b is the element in the top row, and operations are carried out modulo 24. alb 0 4 8 12 16 20 0 4 8 12 16 20 (iii) Show that S is closed under multiplication by completing the following table, where the body of the table contains "ab", where a is the element in the left column, b is the element in the top row, and operations are carried out modulo 24. alb 0 4. 8 12 16 20 0 4