Continuity and Limits of Functions Exercise Answers
1. Let f be given by f (x) =
4 x for x 4 and let g be given by g (x) = x2 for all
x R.
(a) dom(f + g ) = dom(f g ) = (, 4], dom(f g ) = [2, 2] and dom(g f ) =
(, 4]
(b) (f g )(0) = 2, (g f )(0) = 4, (f g
Equality Constraints and the Theorem of Lagrange
Exercise Answers
All the referenece are to exercises from Simon & Blume.
1. Exercises 18.1-18.9.
2. Exercise 19.14.
3. Find the maximum and minimum of f (x, y ) = x2 y 2 on the unit circle x2 + y 2 = 1
usin
Functions and Cardinality Exercise Answers
1. (a) A function.
(d) Not a function.
(b) Not a function.
(e) Not a function.
(c) A function
(f) A function.
2. (a) The domain and codomain are not specied, nor does it say for what x the
statement f (x) = cos x
Implicit Functions Exercise Answers
All the exercises are from Simon & Blume.
1. Exercise 15.1
Solution.
(a) We have to check that the conditions of the implicit function theorem are satised. We have
G(x, y ) = x2 xy 3 + y 5
which is continuously dierenti
Linear Independence Exercise Answers
1. Exercise 11.1.
2. Exercise 11.2. Answers in S&B.
(a) Let 1 and be scalars. To use the denition of linear independence directly,
consider the equation
1
2
1
+ 2
1
2
21 + 2
1 + 22
=
=
0
0
.
Thus we need 21 + 2 = 0 and
Systems of Linear Equations Exercise Answers
1. Exercise 7.11.
(a) First write the system in matrix form, using an augmented matrix.
3
3| 4
1 1 | 10
Interchange rows 1 and 2 (to avoid having to use fractions and add 1/3 of row
1 to row 2 to eliminate the
Metric Space Properties of Euclidean Spaces Exercise
Answers
All the exercises are from Simon & Blume.
1. Exercise 12.11. Show that if cfw_xn and cfw_yn are two sequences of vectors in
n=1
n=1
k
R convergent to x and y respectively, then cfw_xn yn conv
Multivariable Exercise Answers
All the exercises are from Simon & Blume.
1. Any of Exercises 13.1-13.10.
2. Exercises 13.11-13.14.
3. Exercise 13.19. Let f = (f1 , . . . , fm ) be a function from Rk to Rm . Then, f is
continuous at x i each of its compone