unit 4 Flashcards
x = — is a critical point of f because
f’ = 0 or is undefined
x = — is a relative MAXimum of f because
f’ changes from positive to negative
+ –> -
POS to NEG
x = — is a relative MIN of f because
f’ changes from negative to positive
- –> +
NEG to POS
x = — is neither a max nor min because
f’ does not changes signs
f is increasing on ( , ) because
f’ > 0
above x-axis
f is decreasing on ( , ) because
f’ < 0
below x-axis
x = — is a point of inflection because
f’’ (the slope of f’) = 0 or is undefined AND changes signs
f is concave UP on ( , ) because
f’’ (the slope of f’) > 0
+
f is concave DOWN on ( , ) because
f’’ (the slope of f’) < 0
-
if f(x) = integral from x to number of g(t) dt, then
g’(x) = f(x)
steps to find relative extrema
- find derivative
- set equal to 0
- solve for critical points
- make table of values, include interval values, and critical points
> plug in x-values into og function
rel min at — when x = — (smallest y-value)
rel max at — when x — (largest y-value)
how to know if rolle’s theorem applies ?
set f(a) and f(b) equal to each other, if it’s true then rolle’s does apply
f(a) = f(b) then rolle’s applies
how to determine values of c guaranteed by the MVT ?
find derivative of function
set equal to 0, and find c
make sure it’s in the interval !!!
what must be met for mvt to apply ?
function must be continuous and differentiable
how to find critical points ?
find derivative
set equal to 0 and find x
critical points are x
find open intervals on which function is increasing or decreasing
find derivative, set equal to 0
(find critical points)
make number line with critical points
f’ changing from POS -> NEG = MAX
f’ changing from NEG -> POS = MIN
how to find when a function is cc up or cc down ?
find derivative
find second derivative
set 2nd derivative equal to 0, and solve
make number line
function is cc up when it’s POS
function is cc down when it’s NEG
how to find points of inflection of function?
find derivative
find second derivative
set equal to 0 and solve
those values are poi
f has points of inflection at x = — because ?
f’’ changes signs
