93年 - 93 台灣聯合大學系統(清、交、陽、中四校聯招)學士班轉學生聯合招生試題：微積分#113225

1. (7分) Let f(x) be a differentiable function on R satisfying
 for all x R. Then f(π)=__________.

2. (7分) Let L be the line tangent to the polar curve r(θ) = at θ = 0. The equation of L in x and y is_________ .

3.(7分) Evaluate the improper integral by transforming it into a definite integral of the form via an appropriate 1-1 onto differ-entiable function . 

4.(7分) Evaluate dx=_____.

5. (8分) Let p(x)=x⁶+2x⁵-x+1. Find =_________·

6.(8分) Evaluate S Sn xydady, where Ωis the region in the first quadrant bounded by the curves:x² + y² =4,x²+y² =9,x² -y² =1,x²-y²=4. 

7.(8分) Evaluate the line integraldy along the path C: y = tan(x) from x = 0 to x = .

1.(10分) Find

2. (10分) Let 0.a₁a₂a₃a₄ ...be the decimal expansion of the rational number . Let b_k = a2k,h = 1,2,...The decimal 0.b₁b₂b₃b₄... also represents a rational number . Find .

3.(10分) Find the shortest distance from the point (1,2,0) to the elliptic cone z =

4.(10分) Evaluate the surface integral f ∫ ∫s(x⁴ + y⁴+ z⁴)dσ, where do is the surface element and S ={ (x, y, Z) : x² + y² + z² =1}