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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
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1 2
n n
n n n
∑ + +
n
n n
x n
∑ ⋅
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.
a) the length of the curve for 0 2
b) the unit tangent vector T
at 2
d) the principal unit normal vector N
at 2
e) equations for the tangent line at 2
2 2 2 2 2
z z z 1 z
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.
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.
D
S
Useful information ( )
b
a
T t r^ t r t
N t T^ t T t
G B ( )^ t^^ =^ T t ( )^^ × N t ( )
(^3 2 )
dT T t^ r^ t^ r^ t^ f^ x ds r t (^) r t (^) f x
a v r^ t^ r^ t r t
a v r^ t^ r^ t r t
Answers
4 0
n (^) n
n
x n
(^1 4 )
1
n (^) n
n
x n
4a.^5 2
− 4c. 3 25
, but
2 0 3 2 0 lim 7 lim 7 7 x x 1 y x x → (^) x x → x
; 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
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)