Download OceanicCalculus1012Exams (2)

國立中山大學
101 學年度第二學期
海工系/海洋學程
102.4.26
微積分(二) 期中考
Attempt ALL the problems. Part I has 13 blanks each of which carries 5 points.
Show the details of your work for Part II, which carries 40 points. The total is 105.
I. Fill in the blank.
1.
R∞
2
e−x dx =
.
2. Determine the convergence/divergence of the following series. Give simple
reasons.
(a)
∞
X
1
;
n3
n!
(b)
n
n=1 n
n=1
∞
X
.
3. For the function f (x) = 4x2 + x − 1, the Taylor polynomial of order 3 at
x0 = 0 is given by P2 (x) =
.
4. The Maclaurin series for the function ex is
.
5. The first three terms of the Taylor series for f (x) = (1 + x)5/2 at x =
1
2
is
.
6. Let f (x, y) = xey + sin(xy). Then fx =
; fxy =
7. For the above problem, the gradient vector ∇f (π, 1) =
.
. And
when u = ( √12 , √12 ), the directional derivative Du f (π, 1) =
. One direction v ∈ R2 such that Dv f (π, 1) is maximum is given by
v=
.
8. Let w = ln(x2 + y 2 ) with x = u sin v and y = uev . When (u, v) = (3, 0),
∂w
the partial derivative
=
.
∂u
9. For the above function w = g(u, v), the tangent plane at (u, v) = (3, 0) is
.
1
II. Show details of your work.
10. (10%) Find the interval of convergence for the power series
∞
X
xn
.
n=0 (n + 1)!
11. (15%)
(a) Find the Maclaurin series for the function
1
.
1+x
(b) Integrate the above series to give the Maclaurin series for ln(1 + x).
(c) Hence or otherwise, express ln 2 in terms of a series.
12. (10%) Let f (x, y) = 2x2 + 2xy + y 2 − 2x.
(a) Find all the critical points of the function.
(b) Determine whether each critical point is a relative maximum, relative
minimum or saddle point.
13. (5%) Consider the function
f (x, y) =



xy
x2 +y 2

 0
(x, y) 6= (0, 0)
.
(x, y) = (0, 0)
Is f continuous at every point (x0 , y0 ) ∈ R2 ? Give argument for your
answer.
End of Paper
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國立中山大學
101 學年度第二學期
海工系/海洋學程
102.6.21
微積分(二) 期末考
Attempt ALL the problems. The total is 102.
I. True or False (T/F) (3%)
1(a) If a function f : Rn → R is differentiable at x0 ∈ Rn , then it is continuous
at x0 . If f is continuous at x0 , then it has limit at x0 .
(b) Every function can have a Taylor series at some point x0 in its domain.
(c) The gradient vector ∇f (x) is a vector which is tangent to the level curve
f (x) = c for some constant c.
II. Fill in the blank. (52%)
2. Determine the convergence/divergence of the following series. Give simple
reasons.
∞
X
n
+1
n=1
∞
X n!
(b)
n
n=1 10
(a)
;
n2
.
3. The Maclaurin series for the function e3x is
.
4. The Taylor polynomial of order 3 for f (x) = (1 + x)5/2 at x0 = 0, P3 (x) =
.
5. Let f (x, y) = x2 y + xy . Then ∇f (x, y) =
, and fxxx =
.
6. For the above problem, when u = ( 35 , 45 ), the directional derivative
Du f (2, 1) =
.
7. Let w = ln(x2 + y 2 ) with x = 2uv and y = u2 − v 2 . For the above function
w = g(u, v), the tangent plane at (u, v) = (3, 0) is
8. Let R = [0, 2]×[0, 1], then the double integral
ZZ
R
.
1
xy 2 dA =
.
9. Let D is the solid sphere x2 + y 2 + z 2 ≤ 1, then
ZZZ
√
D
x2
1
dV =
+ y2 + z2
.
10. For the transformation x = u2 − v 2 , y = 2uv for a double integral, then the area elements have the ratio dxdy = g(u, v)dudv, where
.
g(u, v) =
11. Let C be the curve {(t, t2 ) : t ∈ [1, 3]}.. The line integral
Z
xy −2 dx +
C
yx−2 dy =
.
12. Let S be the surface of the unit sphere x2 +y 2 +z 2 = 1 such that x, y, z ≥ 0.
ZZ
The surface integral
z 2 dS =
.
C
III. Show details of your work. (47%)
13. (a) Find the Maclaurin series for the function
1
.
1−x
(b) Integrate the above series to give the Maclaurin series for ln(1 − x).
(c) Hence or otherwise, express ln 2 in terms of a series.
14. Find the maximum and minimum values of the function f (x, y, z) = x −
y + z on the sphere x2 + y 2 + z 2 = 100.
15. Evaluate the following double integrals.
(a)
ZZ
xy dA where R is the triangle with endpoints (0, 0), (2, 0) and (1, 3).
R
(b) Find the volume of the region that is bounded above by the elliptic
x2
x2
+ y, on the sides by the cylinder
+ y 2 = 1 and
paraboloid z =
9
9
below by the xy-plane.
16. (a) Evaluate
R
C
x dx+y dy
,
x2 +y 2
where C is the unit circle traversed counterclockwise.
(b) Show that if C is any closed curve, then
Z
(x − 3y) dx + (2x − y 2 ) dy = 5A ,
C
where A is the area of the region D enclosed by C.
Hint: Use Green’s Theorem.
2
17. (7%)
(a) Prove the following Taylor expansion theorem at x0 = 0.
1 00
1 000
1 Z x 0000
2
3
f (x) = f (0) + f (0)x + f (0)x + f (0)x +
f (t)(x − t)3 dt .
2
3!
3! 0
0
1 Z x 0000
Hint: Integrate by parts for the term R =
f (t)(x − t)3 dt.
3! 0
(b) Is this R exactly the same as the remainder term in Taylor expansion?
Why?
End of Paper
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