Functions of One Variable

Question 1

How do you check a function is differentiable at xo?

Answer 1

Check limit exists as x -> xo-, x -> xo+ and that xo- = xo+

Question 2

How do you check a function is continuous and differentiable?

Answer 2

Cont: check L- = L+
Diff: check L-, L+ as h -> 0 (use diff definition)

Question 3

What is the equation for the derivative of f(x)?

Answer 3

f’(x) = lim h -> 0 {(f(x+h) - f(x))/h}

Question 4

What is Rolle’s Thm?

Answer 4

If f(x) is cont on the closed interval [a,b] and diff on the open interval (a,b) with f(a) = f(b) then there exists at least one point c, st:

f’(c) = 0

Question 5

What is the Mean Value Thm?

Answer 5

If f(x) is cont on [a,b] and diff on (a,b) then there exists c st:

f’(c) = f(b) - f(a)/b-a

Question 6

What is the exp form of sinh and cosh?

Answer 6

sinhx = 1/2(e^x - e^(-x))
coshx = 1/2(e^x + e^(-x))