理工 A 1
國立聯合大學九十九學年度第一學期
理工學院微積分第二次會考試題 A 卷
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學號：
一 .單 選 題 (1—15 每 題 4 分 )
5x − 2x
.
x→0
x
(A) ln 5 (B) ln 5 − ln 2
1. Find lim
(D) 5x ln 5 − 2x ln 2
(C) ln 3
(E) 1
2. If f (x) = x, g(x) = ln x, which of the following limits is an indeterminate form ?
g(x)
(C) lim f (x)g(x) (D) lim [f (x) + g(x)]
(A) lim+ f (x)g(x) (B) lim+
x→∞
x→∞
x→0
x→0 f (x)
√
3. If y = ln 3 1 + 2x, find y ′ .
(A) √
3
1
1 + 2x
(B)
3
(1 + 2x)
(C) √
3
2
1 + 2x
(D)
2
3(1 + 2x)
(E)
g(x)
x→1 f (x)
(E) lim
2
3( 1 + 2x)2
√
3
4. If y = x ln 3, find y ′.
(A) ln 3
(B) ln 3 +
x
3
(C) 0
(D)
x
3
(E) ln x
5. If y = xx , find y ′.
(A) x · xx−1
(B) 1
(C) xx ln x
(E) xx (1 + ln x)
(D) 1 + ln x
6. If f (x) = e2x , find f (10) (x).
(A) 2x(2x − 1) · · · (2x − 9)e2x−10
Z
7. Find tan x dx.
(C) 210 e2x
(B) 2x10 e2x
(E) 2e2x
(D) 0
(A) sec2 x + C
(B) ln | sec x| + C
(C) cot x + C
(D) ln | cos x| + C
(E) tan x sec x + C
8. If f (x) = x|x| , which of the following statements is NOT correct?
(A) f ′′ (0) does not exist.
(B) (0, 0) is the infletion point of the curve y = f (x).
(C) f has no local extreme value.
(D) f has a critical number x = 0.
(E) f ′ (0) does not exist.
9. For the function f ′′ whose graph is given, find the x- coordinates (x-座標 ) of the inflection points.
(A) 1,4,7
(B) 3,5
(C) 1, 7
(D) 2,4,6 (E) 2,6
√
10. Evaluate the Riemann sum for f (x) = x3 + 1, −1 ≤ x ≤ 3 with four equal subintervals, taking the sample points to be
left endpoints.
√
√
√
√
√
(A) 4 + 2 + 28 (B) 4 2 (C) 4 (D) 4 + 2 (E) 2
Z 5
Z 5
11. For the function f whose graph is given, let A =
f (x) dx, and B =
|f (x)| dx, then
1
第 11 題未完，下頁繼續
1
理工 A 2
繼續上頁第 11 題
(A) A = 3, B = 7
(B) A = 7, B = 3 (C) A = 2, B = 12 (D) A = 12, B = 2
Z 3√
9 − x2 dx. (Hint: By interpreting it in terms of areas)
12. Evaluate the integral
(E) A = 7, B = 7
−3
(A) 18
(B) 9π
(C) 9
(D) 3
(E)
9
π
2
1
x+1
dx.
2
0 x +1
1
1
π
π
1
(A) ln 2 (B) ln 2 (C) ln 2 −
(D) ln 2 +
2
2
4
2
4
Z 3
14. Let F (x) =
t2 sin t dt, the derivative of F (x) is
13. Evaluate the integral
Z
(E)
π
4
0
2
(A) x sin x
(B) 9 sin 3
(C) 0
(D) 3
(E) 2x cos x
15. Find the equation of the tangent line to the curve x − x3 y + 5 = y 2 at point (1, 2).
(A) y = 2
(B) x + y = 3
(C) 5x + 3y = 11
(D) x − 7y = −13
(E) x + 7y = 15
二 .複 選 題 (16—19 每 題 5 分 )
16. For the function f whose graph is given. Let g(x) =
Z
x
f (t) dt, which of the following statements are correct?
0
(A) g(1) < g(3)
(B) g(x) has a local maximum at x = 5.
(C) g(3) < g(5)
(D) g(x) has the abosolute maximum at x = 9.
(E) g(7) = g(3)
n
X
i3
as a definite integral and find the value, which of the following statements are correct?
4
n→∞
n
i=1
17. Express A = lim
1
4
1
(B) A =
5
Z 1
(C) A =
x3 dx
0
Z n
(D) A =
x4 dx
(A) A =
1
第 17 題未完，下頁繼續
理工 A 3
繼續上頁第 17 題
(E) A is the limit of Riemann sum for f (x) = x3 , 0 ≤ x ≤ 1, with n equal subintervals, taking the sample
points to be right endpoints.
18. For the function f whose graph is given, which of the following statements are NOT correct?
(A) f is concave upward on the interval (0, 3).
(B) f has local minimum values at x = 1 and x = 4.
(C) f has two critical numbers.
(D) f has a local maximum value f (6).
(E) x = 2 is an inflection point of the curve y = f (x).
x2
, which of the following statements are correct?
x2 + 3
(A) The domain of f is (−∞, ∞).
19. If f (x) =
(B) f has three critical numbers.
(C) f has a local maximun value at x = 0.
(D) There are two inflection points to the curve y = f (x).
(E) f is concave upward on the interval (−1, 1).
三 .單 選 題 (20—25 每 題 5 分 )
1
20. Find lim+ xe x .
x→0
(A) 0
(B) 1
(A) π
(B) ∞
(C) ∞
sin x
−1
.
21. Find lim sin
x→0
x
(C) 0
(D) e
(E) 2
(D) does not exist
(E)
√
22. If f (x) = (tan−1 x)3 , find f ′ (x).
√
1
√
(A) 3(tan−1 x)2 ·
2(1 + x) x
√
1
(B) 3(tan−1 x)2 ·
(1 + x2 )
√
1
√
(C) 3(tan−1 x) ·
2(1 + x) x
√
1
(D) 3(tan−1 x)2 ·
(1 + x)
√
1
(E) 3(tan−1 x)2 · √
2 x
Z
Z
e2x
x
dx = f (u) du, then
23. Let u = e ,
e2x + 3ex + 2
(A) dx = du
u
(B) f (u) = 2
u + 3u + 2
2u
(C) f (u) =
5u + 2
u2
(D) f (u) = 2
u + 3u + 2
x
(E) dx = e du
π
2
理工 A 4
24. Find
Z
(e−x + ex )2 dx.
2
2
(A) e−x + ex + 2x + C
(B) e−2x + e2x + 2x + C
(e−x + ex )3
+C
3
(e−x + ex )3
+C
(D)
3ex
1
1
(E) − e−2x + e2x + 2x + C
2
2
Z 1
√
25. Find
x 1 − x dx.
(C)
0
(A) −
4
15
(B) −
2
3
(C)
2
3
(D)
4
15
(E)
16
15