Final examination Calculus III 201-BZF-05 December 2005
1. (6 points) Determine whether the series is absolutely convergent, conditionally convergent or divergent:
2
1
52
(1)673
n
n
n
nn
∞
=
+
−
+
+
∑
__________________________________________________________________________________________
2. (6 points) Determine the interval of convergence for the following power series:
()
1
3
2
n
n
n
x
n
∞
=
−
⋅
∑
__________________________________________________________________________________________
3. (9 points) a) Use the power series for
()
()
24 2
0
...
cos 1 1
2! 4! 2 !
n
n
n
x
xx
xn
∞
=
=− + = −
∑
to find a power series for
(
)
2
cos
x
.
b) Use this series to find a series for
(
)
2
2
1cos
()
x
fx
x
−
=
c) Calculate
1
0
()
f
xdx
∫with an error less than 0.005.
__________________________________________________________________________________________
4. (10 points) For () 4,3cos,9 3sin 0rt t t t t=−≥
G, find each of the following:
a) the length of the curve for 02
t
π
≤
≤.
b) the unit tangent vector T
J
G at 2
t
π
=
.
c) the curvature
κ
at 2
t
π
=.
d) the principal unit normal vector N
J
JG at 2
t
π
=
.
e) equations for the tangent line at 2
t
π
=
.
__________________________________________________________________________________________
5. (6 points) Show that 32
(,) (0,0)
7
lim
xy
x
y
x
y
→+does not exist.
__________________________________________________________________________________________
6. (9 points) Given that (, )zfxy=, cosxr
θ
= and sinyr
θ
=
, show that
2
222
2
1zzz z
xyrr
θ
⎛⎞
∂∂∂ ∂
⎛⎞ ⎛⎞⎛⎞⎛ ⎞
+=+
⎜⎟ ⎜⎟⎜⎟⎜ ⎟
⎜⎟
∂∂∂ ∂
⎝⎠ ⎝⎠⎝⎠⎝ ⎠
⎝⎠ .
__________________________________________________________________________________________
7. (9 points) Given the function ( , , ) z
f
x y z xye=, the point
(
)
2,4,0P, and the vector 2 2vijk=+ +
GGG
G:
a) Find the directional derivative of ( , , )
f
xyz at the point P in the direction of v
G
.
b) What is the maximum rate of increase of ( , , )
f
xyz at the point P?
c) Find equations for the plane tangent to the level surface ( , , ) 17fxyz= at P.
__________________________________________________________________________________________
8. (9 points) Find and classify all critical points of 33
(, ) 3 12 7fxy x y x y
=
+−− +.
__________________________________________________________________________________________
