Mathematics of Engineering II
SS 2016
Prof. Dr. R. Reemtsen
M. Sc. M. Bähr
Exercise Sheet 11 - Solution
Problem 11.1
Problem 11.2
Consider each sequence separately
1+k
2
k→∞ k
a) lim xk1 = lim
k→∞
= 0 and lim xk2 = lim k +
k→∞
k→∞
1
k
k
b) lim xk1 = lim 1+ k1 = e1 and lim xk2 = lim
k→∞
k→∞ k→∞
k→∞
e1
with lim ~xk =
0
k→∞
= ∞, therefore ~xk is divergent
1 k
2
= 0, therefore ~xk is convergent
Problem 11.3
y3
=0
y→0 y 2
lim f (0, y) = lim
y→0
lim f (x, 0) = lim
x→0
x→0
0
=0
x2
t2 + t3
1+t
1
= lim
= 6= f (0, 0)
2
2
t→0 t + t
t→0
2
2
lim f (t, t) = lim
t→0
therefore f is not continuous at (0, 0)
Problem 11.4

a)
2
(2x + 3) ln(y + 3)

∇f = 
2y(x2 +3x)
y 2 +3




H=

3y 2
x4 


− 2y
x3

H=

∇f = 
(2x+3)2y
y 2 +3
2
2 ln(y + 3)
− x63 +

b)

18
 x4
(x2 +3x)(6−2y 2 )
(y 2 +3)2
(2x+3)2y
y 2 +3
−
6y
x4
12y 2
x5
6y
x4
− x23








c)
z3 y2
x







3
∇f =  2yz ln(x) 




3z 2 y 2 ln(x)

d)

∇f = Q~x + ~b = 
3 2

− zxy2


 z 3 2y
H=
 x


x1 + 2x2 − 1
2x1 + 4x2 + 4
3z 2 y 2
x
z 3 2y
x
2z 3 ln(x)
3z 2 y 2
x



2
6yz ln(x)



6yz 2 ln(x) 6zy 2 ln(x)




1 2
H=Q=
2 4
!
Homework
(due April 13/18, 2016)
Problem H 11.5
Problem H 11.6 Consider
2
=0
k→∞ k
lim xk1 = lim
k→∞
4k − 3
=2
k→∞ 2k + 10
lim xk2 = lim
k→∞
k2
=0
k→∞ k 3
lim xk3 = lim
k→∞
(−1)k k
= divergent
k→∞ k + 1
lim xk4 = lim
k→∞
therefore ~xk is divergent
Problem H 11.7
y2
0
=0
+2
x→0 x2
0
=0
+2
lim f (0, y) = lim
y→0
y→0
lim f (x, 0) = lim
x→0
t2
= 0 = f (0, 0)
t→0 t2 + t2 + 2
lim f (t, t) = lim
t→0
therefore f is continuous at (0, 0)
Problem H 11.8

eyx+3x (y + 3)
a)

∇f = 

b)

∇f = 

c)
eyx+3x x





H=
2 sin(2x) − 2xe5y
−5e5y x2 + 6y
3x2 y 5 z 2





 3 4 2
4
∇f = 5x y z + z 2 




2x3 y 5 z −
8y
z3

eyx+3x (y + 3)2
eyx+3x (x(y + 3) + 1)
eyx+3x (x(y + 3) + 1)





H=

eyx+3x x2
−4 cos(2x) − 2e5y
6xy 5 z 2



H = 15x2 y 4 z 2


6x2 y 5 z
−10xe5y
−10xe5y
−25e5y x2 + 6
15x2 y 4 z 2
6x2 y 5 z
20x3 y 3 z 2
10x3 y 4 z −


8
z3






8
3 4
10x y z − z 3 


2x3 y 5 +
24y
z4