theorems review !!

Question 1

squeeze theorem

Answer 1

REQ: h(x) ≤ f(x) ≤ g(x)
lim as x -> 0 h(x) = L = lim as x -> c g(x)

GUAR: lim as x -> c f(x) = L
^^ this guarantees a limit value

Question 2

Intermediate Value Theorem

Answer 2

REQ: continuity on [a, b]

GUAR: a ‘c’ on (a, b) where f(c) is btwn f(b) & f(a)

Question 3

extreme value theorem

Answer 3

REQ: f is continious on [a, b]

GUAR: one ABS. max & one ABS. min

Question 4

fundamental theorem of calculus

Answer 4

REQ: continuity on [a, b]
f is an ANTI-derivative of f

GUAR: integral from a to b (b on top) of f(x) dx equals F(b) - F(a)

integral from a to b (b on top) of f PRIME (x) dx equals f(b) - f(a)

Question 5

2nd fundamental theorem of calculus

Answer 5

REQ: continuity on an integral containing ‘a’

GUAR: d/dx of integral from g(x) to a (g(x) on top) of f(t) dt = f(g(x)) * g’(x)

Question 6

average value of a function

Answer 6

REQ: continuity on [a, b]

GUAR: avg. value of f on [a, b] = 1/ (b-a) integral from b to a (b on top) of f(x) dx

Question 7

mean value theorem

Answer 7

REQ: continuity on [a, b]
AND differentiablility on (a, b)

GUAR: a ‘c’ on (a, b) such that f’(c) = f(b) - f(a) / b - a

Question 8

rolle’s theorem

Answer 8

REQ: f is continious on [a, b]
AND f is differentiable on (a, b)
f(b) = f(a)

GUAR: a ‘c’ on (a, b) where f’(c) = 0