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Calculus III Final Examination December 2005, Exams of Calculus

The final examination questions for calculus iii held in december 2005. The questions cover topics such as series convergence, power series, integration, and calculus of functions of several variables.

Typology: Exams

2012/2013

Uploaded on 02/12/2013

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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
=
+−− +.
__________________________________________________________________________________________
pf2

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Final examination Calculus III 201-BZF-05 December 2005

  1. (6 points) Determine whether the series is absolutely convergent, conditionally convergent or divergent:

1 2

n n

n n n

∑ + +


  1. (6 points) Determine the interval of convergence for the following power series: (^ ) 1

n

n n

x n

∑ ⋅


  1. (9 points) a) Use the power series for (^) ( ) ( )

2 4 2 0

cos 1 ... 1 2! 4! 2!

n^ n n

x x^ x^ x n

= − + = (^) ∑ −

to find a power series for cos (^) ( x^2 (^) ).

b) Use this series to find a series for (^ )

2 2

1 cos ( )

x f x x

c) Calculate

1

0

f^ ( ) x^ dx with an error less than 0.005.


4. (10 points) For r t^ G^ ( ) = 4 , 3cos , 9 t t − 3sin t t ≥ 0 , find each of the following:

a) the length of the curve for 0 2

≤ t ≤ π.

b) the unit tangent vector T

JG

at 2

t = π.

c) the curvature κ at

t = π.

d) the principal unit normal vector N

JJG

at 2

t = π.

e) equations for the tangent line at 2

t = π.

__________________________________________________________________________________________

  1. (6 points) Show that (^) ( x y , lim) (0,0) 37 xy 2 → (^) x + y does not exist.

6. (9 points) Given that z = f ( , x y ), x = r cos θ and y = r sin θ, show that

2 2 2 2 2

z z z 1 z

x y r r θ

⎛ ∂^ ⎞ + ⎛^ ∂^ ⎞ = ⎛ ∂^ ⎞ +⎛ ⎞⎛ ∂ ⎞

__________________________________________________________________________________________

  1. (9 points) Given the function f ( , x y z , ) = xyez , the point P (^) ( 2, 4,0), and the vector v = 2 i + 2 j + k

G G^ G^ G

a) Find the directional derivative of f ( , x y z , ) at the point P in the direction of v^ G^. b) What is the maximum rate of increase of f ( , x y z , )at the point P? c) Find equations for the plane tangent to the level surface f ( , x y z , ) = 17 at P.


  1. (9 points) Find and classify all critical points of f ( , x y ) = x^3^ + y^3 − 3 x − 12 y + 7.

Final examination 201-BZF-05 December 2005 page 2

9 (9 points) A box is to have all of its vertices on the ellipsoid

2 2 2 1 9 4

x (^) + y + z =. Find the dimensions of the

box with maximum volume.


  1. (9 points) Calculate the volume of the solid bounded below by the xy plane, above by z = 49 − x^2 , and on the sides by x = 0, y = 2 x and y = 6 − x ..

11. (9 points) D is the region in the first quadrant inside the circle r = 4 and outside the circle r = 4 cos θ.

Assume that the surface density δ is constant. Calculate the moment about the y axis: y

D

M = ∫∫ δ x dA.

__________________________________________________________________________________________

  1. (9 points) S is the solid in the first octant bounded by the cylinder x^2^ + y^2 = 4 and the sphere

x^2^ + y^2 + z^2 = 25. Assume that the density δ is constant. Calculate the moment of inertia about the z axis:

z^ (^2 2 )

S

I = ∫∫∫ δ x + y dV.

__________________________________________________________________________________________

Useful information ( )

b

a

s = ∫ r G ′^ t dt ( ) ( )

T t r^ t r t

G^ G

G

N t T^ t T t

G G

G B ( )^ t^^ =^ T t ( )^^ × N t ( )

G G G

( (^ ) )

(^3 2 )

( ) ( )^ ( )^ ( )

dT T t^ r^ t^ r^ t^ f^ x ds r t (^) r t (^) f x

′ ′^ × ′′^ ′′

G G G G

G G

T ( )

a v r^ t^ r^ t r t

= ′= ′^ ⋅ ′′

G G

G 2

N ( )

a v r^ t^ r^ t r t

′ × ′′

G G

G

cos^2 1 ( 1 cos 2 )

θ= + θ cos 4 1 3 2 cos 2( ) 1 cos 4( )

θ= ⎛⎜^ + θ + θ ⎞⎟

sin^2 1 ( 1 cos 2 )

θ= − θ sin 4 1 3 2 cos 2( ) 1 cos 4( )

θ= ⎛⎜^ − θ + θ ⎞⎟

__________________________________________________________________________________________

Answers

1. conditionally convergent 2. 1 ≤ x < 5 3a. (^ )

4 0

n (^) n

n

x n

∑ 3b.^

(^1 4 )

1

n (^) n

n

x n

∞^ +^ −

∑ 3c. 0.

4a.^5 2

π 4b. 4 , 3 , 0

− 4c. 3 25

4d. 0, 0,1 4e. x = 2 π+ 4 , t y = −3 , t z = 6

  1. 0 lim 0 7 (0) 3 0 x 0 y xx

, but

2 0 3 2 0 lim 7 lim 7 7 x x 1 y x x → (^) x xx

; hence

2 0 3 2 lim^7 x

x → (^) x + x does not exist

(^2 2 2 2 2 22 ) 2 cos sin 1 sin cos sin cos sin cos

RHS = f x θ+ f y θ + r f x − r θ + f ry θ = f x θ + θ + fy θ + θ

7a.^20 3

7b. 2 21 7c. 2 x + y + 4 z = 8 8. local min: (1,2) local max: (-1,-2) saddlepoints: (-1,2) (1,-2)

  1. 2 3 x 2 3 x^4 3 3

δ⎛⎜^ − π⎞⎟