Proofs - 2nd Partial

Question 1

Derivative of a constant (H)

Answer 1

Question 2

Derivative of y = xa (H)

Answer 2

Question 3

Derivative of y = ex (H)

Answer 3

Question 4

Derivative of y = lnx (H)

Answer 4

Question 5

Derivative of a linear combination (*)

Answer 5

Question 6

Relationship between derivability and continuity (*)

Answer 6

Question 7

One-variable necessary condition for local maximizers/minimizers (Fermat’s theorem) (H)

Answer 7

Question 8

Rolle’s theorem (H)

Answer 8

Question 9

Lagrange’s mean value theorem (H)

Answer 9

Proof Summary: take g(x), use rolles

Question 10

Characterization of functions with a null derivative (H)

Answer 10

Question 11

Characterization of functions with the same derivative (H)

Answer 11

Question 12

(Strict) Monotonicity test on an interval (*) (2 proofs in 1)

Answer 12

Question 13

N-variable necessary condition for unconstrained local maximizers/minimizers (Fermat’s theorem) (H)

Answer 13

Question 14

Characterization of SpanS with linear combinations (*)

Answer 14

Question 15

Property of unique writing for a basis (*)

Answer 15

Question 16

Determination of the coefficients of a vector in Rn, with respect to an orthonormal basis (H)

Answer 16

Question 17

Riesz’s representation theorem for functions f: Rn -> R(m) (*)
(2 proofs in 1)

Answer 17

Question 18

The image space is a subspace, spanned by the image of the standard basis (*)

Answer 18

Take f:R^n -> R^m Linear

Question 19

The kernel is a subspace (H)

Answer 19

Take f:R^n ->R^m Linear
Note! : Ker f not necessarily a subspace of R^m (if n>m, certainly not)

Question 20

Determinant of the inverse matrix (H)

Answer 20

Question 21

Kronecker-Capelli’s theorem (*)

Answer 21

Question 22

Cramer’s theorem (H)

Answer 22

Question 23

Derivative of y = arctanx (H).

Answer 23

Question 24

Relationship between derivability and differentiability (H)

Answer 24

Question 25

The intersection of subspaces is a subspace (*). 

Answer 25

Question 26

Maclaurin's expansion for y = ex (H)

Answer 26

Question 27

Maclaurin's expansion for y = sinx (H)

Answer 27

Question 28

Maclaurin's expansion for y = cosx (H)

Answer 28

Question 29

Maclaurin's expansion for y = ln(1+x) (H)

Answer 29

Question 30

Maclaurin Expansion y=(1+x)^a (Proof not required)

Answer 30