MIDTERM 2 FOR CALCULUS
Time: 8:10–9:55 AM, Friday, December 15, 2000 Instructor: Shu-Yen Pan
No calculator is allowed. No credit will be given for an answer without reasoning.
1. Find
(1) [4%]R 1
x2−4x+3dx.
(2) [4%]R 1
x2−4x+5dx.
2. (1) [4%] EvaluateR1
−2|2x + 1| dx.
(2) [4%] IntegrateR
tan4x sec4x dx 3. (1) [4%] Find f0 if f (x) = (x2)x.
(2) [4%] IntegrateR
coth x dx.
4. (1) [4%] Find the exact value of the expression sin(cos−1 35).
(2) [4%] Find the area of the region bounded by the curves y = 20 − x2 and y = x2− 12.
5. [6%] Use Newton method with the specified initial approximation x1 = −1 to find x2, the second approximation to the root of the equation x3+ x + 1 = 0.
6. [6%] Suppose that f is differentiable, f (0) = 0, f (1) = 1, f0(x) > 0 and R1
0 f (x) dx = 14. Find the value of the integralR1
0 f−1(y) dy.
7. [8%] Let
f (x) = Z x
2
p1 + t2dt.
Prove that f (x) has an inverse and find f0(0).
8. [8%] Find the integralR2
0 x2 (x2+4)2dx.
9. [8%] Find the limit
n→∞lim 1 n
Ãr1 n+
r2 n+
r3
n+ · · · + rn
n
! .
10. [8%] A fence 3 meters tall runs parallel to a tall building at a distance of 2 meters from the building.
What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
11. [8%] If x sin x =Rx2
0 f (t) dt, where f is a continuous function, find f (4).
12. [8%] Find the limit
x→∞lim µ
1 + 3 x
¶2x .
13. [8%] Let f be a function such that f0 is continuous on [a, b]. Prove that Z b
a
f (t)f0(t) dt = 1 2
³
f (b) − f (a)´³
f (b) + f (a)´ .
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