Week 4

Question 1

At a point on an open interval, define the derivative

Answer 1

Question 2

Answer 2

Question 3

Requirement for derivative (limits)

Answer 3

Question 4

Answer 4

Question 5

Answer 5

Question 6

The set of all continuously differentiable functions on (a,b)

Answer 6

Question 7

Answer 7

Question 8

Answer 8

Question 9

Answer 9

Question 10

Answer 10

Question 11

Answer 11

Question 12

Answer 12

Question 13

Answer 13

Question 14

Answer 14

Question 15

Answer 15

Question 16

Answer 16

Question 17

Answer 17

Question 18

Right derivative of f at point x

Answer 18

Question 19

The left derivative of f at point x

Answer 19

(The one to the right of ‘and’)

Question 20

Quotient rule

Answer 20

Where f and g are differentiable functions

Question 21

Corollary to Fermat’s theorem? Give example

Answer 21

F does not have to be differentiable at the point where it has a local extremum.
Eg:
f(x) = |x| attains (global) min at x=0
But f’(0) doesn’t exist

Question 22

Taylor expansion of exp(x)

Answer 22

Question 23

Taylor expansion of sin(x)

Answer 23

Question 24

Taylor expansion of cos(x)

Answer 24

Question 25

Taylor expansion of log(1+x)

Answer 25

Question 26

Asymptotic expansion of f, a function defined in a neighbourhood of x_0 €R

Answer 26

Question 27

Asymptotic expansion of (below) as x -> 0

Answer 27

Question 28

Compute first 3 non trivial terms of Taylor expansion

Answer 28

Question 29

Requirements for Taylor’s formula to be valid

Answer 29

f€cⁿ(a,b) and x_0 must be in (a,b)