Theorems

Question 1

Average Rate of Change (not avg value!)

Answer 1

f(b) - f(a)/b-a

Question 2

Instantenous Rate of Change

Answer 2

f’(c) - slope at a single point

Question 3

Mean Value Theorem (derivative)

Answer 3

If f is continuous on [a,b] and diff on (a,b):
Then, there exists one value c so that
f’(c) = f(b)-f(a)/b-a
aka where the slope of the tangent equals the slope of the secant

Question 4

Average Value

Answer 4

1/b-a integral of f(x) from a to b

Question 5

Intermediate Value Theorem

Answer 5

if f is cont [a, b], and f(a) does not equal f(b) then: f takes on every y-value between f(a) and f(b)
so like if k is any # between f(a) and f(b), then there exists an f(c) = k!
and if f(a) and f(b) are opposite signs, then there is at least one zero inbetween!

Question 6

Extreme Value Theorem

Answer 6

a continous function on [a,b] has both an absolute minimun and an absolute maxima on [a,b]

Question 7

Definition of Continuity

Answer 7

lim x->c+ f(x) = lim x->c- f(x), f(c) exists, and lim x-> c = f(c)

Question 8

Squeeze Theorem

Answer 8

Got 1 function squeezed between 2 others?
Their limit at x = c will be the same for the middle function, as long as it is the same for the upper and lower ones!

Question 9

Mean Value Theorem (integral)

Answer 9

If continuous [a,b], then there’s a value c where
integral of f(x) from a to b = f(c)(b - a)
a continuous function on a closed interval takes on its average value at least once within that interval

Question 10

Start Plus theorem

Answer 10

f(b) = f(a) + integral from a to b f’(x) dx