[試題] 104資工轉系考 微積分
題號可能有錯，還請各位大大幫忙修正><
1. Write down the definition of The Fundemental Theorem of Calculus.
sin x^4
2. (a) lim --------- = ?
x→∞ sin^4 x
cot x
(b) lim (1+x) = ?
x→0+
-1
3. (a) ∫sin (2x) dx = ?
x^2 + 3x + 2
(b) ∫-------------- dx = ?
3x (x^2 + 2)
(3n)! x^n
4. Find the interval of convergence of Σ -----------.
x→∞ n^(3n)
5. Solve the initial value problem dy/dx = sqrt(4-y^2), y(π/3) = 1.
2 2
6. Find and classify the critial points of f(x,y) = x + ?? + xy + ??.
(忘記方程式了)
2 2 2 2
7. Find the points on the intersection of 3x + y = 2 and x + 2y + z = 3
that are closet and furthest from the origin.
(不確定方程式)
2x - y
8. ∫∫ sin (--------) dA, where R is enclosed by 1 ≦ 2x + y ≦ 2,
R 2x + y
2x - y = 0 and 2x + 5y = 0.
2 √x 2 3
9. f(x) - x = ∫ f(t )t dt. Find f(x).
1
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