ap review time babyyyyy Flashcards
define extreme value theorem
if a function is continuous on the CLOSED INTERVAL [a, b] then x must attain a maximum AND a minimum at least once
if a function is differentiable it is also…..
continuous!
if a function is differentiable it is also…..
.
continuous
if a function is differentiable it is also…..
CONTINUOUS
if a function is differentiable it is also…..
CONTINUOUS PLZ PLZ JUSTIFY THIS ON LITERALLY ANY FRQ WHERE ITS LIKE “explain how you got here”
define intermediate value theorem
if function is continuous on a CLOSED INTERVAL [a, b] then f(x) must take all all values between f(a) and f(b) within that interval
evt, ivt, and mvt all require…..
continuity on a CLOSED interval (but mvt requires also differentiability on the OPEN interval)
symbol for therefore (on frq’s)
the trinity of dots (one dot top, two at bottom)
mean value theorem
if f is continuous on the CLOSED interval [a, b] and DIFFERENTIABLE on the OPEN interval (a, b), then there is a point on w/in that open interval which the tangent line is equal to the secant line/avg rate of change of the ENDPOINTS
f’(c) = ( f(b) - f(a) ) / ( b-a )
to find the limit of ANYTHING, it must be….
- continuous OR continuous by being differentiable
- that’s, like, it
linear approxmiation equation
L(x) = f(a) + f’(a)(x-a)
rolles theorem
essentially mvt but with f(a) = f(b) and f’(c) = 0 (horizontal)
if f is continuous on [a, b] and differentiable on (a, b) where f(a) = f(b), there must be a point with that Gucci nice horizontal tangent line
what equation does “the rate of change of a quantity is directly proportional to the size of the quantity” mean?
dy/dx = constant * y, or the differential equation for EXPONENTIAL growth
what is the equation for “the rate of change of a volume over time is proportional to the reciprocal of the cube of the volume”?
break it up into terms,
dv/dt is proportional to 1/v^3
and as all rates of change need some constant, whether it be 1 or anything else,
dv/dt = 1/v^3 * k
or
dv/dt = k/v^3
