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

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:

1.The upper sum = __(A)__. and the lower sum = __(B)__ if f(x) := |x|,x ∈ [-2,1] and P is the partition of [-2,1].

2.=__(C)__

3.=__(D)__

4.=__(E)__ for x＞ 0.

5.The radius of convergence of the power series =__(F)__

6.The Maclaurin serics for sin(x + 1) =__(G)__