109年 - 109 台灣聯合大學系統(清華、交通、陽明、中央四校聯招)學士班轉學生考試：微積分#112193

科目：研究所、轉學考(插大)-微積分 | 年份：109年 | 選擇題數：0 | 申論題數：12

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所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (12)

1. Find the limit:6397c6d283739.jpg
Answer: _________

2. Find the smallest positive (x ＞ 0) infection point of F(x) = 6397c7506c85a.jpgdt.
Answer :____________

3. How many local extreme values does the function 6397c77bc42be.jpghave?
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. Evaluate6397c7f103a6e.jpg dA where R is the unit disk x2+ y2 ≤1.
 Answer :__________________
6. Find the volume between the two spheres: x2 +y2 + z2 = 1, x2 +y2 + z2 =2 and inside the cone z2 = x2+ y2.
Answer :__________

7. Calculate 6397cbe29d3de.jpg dA where R is the interior of the ellipse6397cc3c36a4e.jpg
Answer : ___________

8. Find the area of the region enclosed by the limagon
x= 2 cost- cos2t, y=2sint-sin2t, 0≤t≤2m
Answer :__________
6397ccde5f883.jpg

1. Let6397cd6464cfa.jpgUse 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 series6397ce2b7be71.jpgdiverges or converges conditionally or converges absolutely and give reasons for your answer. (6 points)

b. Show that if 6397cf597d8b5.jpg an converges,then6397cf6f61ec1.jpg converges. (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 6397cfbd94cc9.jpg, 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:
6397cf9e16487.jpg