6. Find the volume between the two spheres: x² +y² + z² = 1, x² +y² + z² =2 and inside the cone z² = x²+ y².
Answer :__________

2. Find the smallest positive (x ＞ 0) infection point of F(x) = dt. Answer :____________

3. How many local extreme values does the function have? Answer :___________

4. Let C be the curve of intersection of the two surfaces  x2 + y2 + z2 = 3 and (x - 2)2 + (y - 2)2+z2 = 3. Find parametric equations of the tangent line to C at P = (1, 1,1). Answer :__________

5. Evaluate dA where R is the unit disk x2+ y2 ≤1. Answer :__________________

7. Calculate dA where R is the interior of the ellipseAnswer : ___________

8. Find the area of the region enclosed by the limagon x= 2 cost- cos2t, y=2sint-sin2t, 0≤t≤2m Answer :__________

1. LetUse polar coordinates to show that f(x, y) is continuous at all (z, y) ifp  ≤ 2, but discontin- uous at (0,0) if p ≤ 2.

a. Dotermine whether the seriesdiverges or converges conditionally or converges absolutely and give reasons for your answer. (6 points)

3. Goods I and 2 are available at prices (in dollars) of p1 per unit of good 1 and p2 per unit of good 2. A utility function U(x1, x2) is a function representing the utility or beneit fo consuming xj, units of good j. The marginal utility of the jth good is , the rate of increase in utility per unit increase in the jth good. Prove the following law of economics: Given a budget of L dollars, utility is maximized at the consumption level (a,b) where the ratio of marginal utility is equal to the ratio of prices:

b. Show that if an converges,then converges. (6 points)