Unit 3 Flashcards
With what do we differentiate with respect to in related rates questions?
time
volume of a sphere
(4/3) · π · r3
surface area of sphere
(4) · π · r2
volume of cone
(1/3) · π · r2 · h
when is x=c a critical number
if f(c) exists and f’(c)=0 or dne
Extreme Value Theorem
if f continuous on a closed interval [a,b], then f has both an absolute minimum and an absolute maximum on the interval
can absolute extremas be endpoints?
yes
3 steps for justifying absolute extrema of continuous function f on [a,b]
- derivative of f
- critical number
- function values evaluated at the endpoints and critical endpoints
Mean Value Theorem
If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c between a and b that
f’(c) = [f(b) -f(a)]/ b-a
conditions for MVT
- continuous [a,b]
- differentiable (a,b)
Rolle’s Theorem
Let f be continuous n the closed interval [a.b] and differentiable on the open interval; (a,b), and f(a)=f(b), then there is at least one c between a and b such that f’(c) = 0
conditions for RT
- continuous [a.b]
- differentiable (a,b)
- f(a)=f(b)
what is happening to f if f’(x)>0 on [a,b]
f is increasing
what is happening to f if f’(x)<0 on [a,b]
f is decreasing
suppose that c is a critical number of a continuous function f, then f has a local maximum at c if
f’ changes from positive to negative
suppose that c is a critical number of a continuous function f, then if has a local minimum at c if
f’ changes from negative to positive
let f be differentiable on an open interval, the graph of f is concave upward on f if
f’ is increasing on the interval
let f be differentiable on an open interval, the graph of f is concave donward on f if
f’ is decreasing on the interval
f is concave up if f”
f”>0
f is concave down if f”
f”<0
inflection point
point on the curve where curve changes concavity
f has a local min if
f’=0 and f’>0
f has a local max if
f’=0 and f’<0
condition for finding local extremes using c
c is a critical number
condition for finding concavity
f is differentiable on an open interval
condition for inflection point
f is continuous
first derivative test
suppose that c is a critical number of a continuous function f
then f has a local maximum at c if f’ changes from positive to negative
then if has a local minimum at c if f’ changes from negative to positive
what happens to then slope of f’ when the graph is concave up? concave down?
increasing; decreasing
even graphs
symmetry at y-axis
f(-x)=f(x)
odd graphs
symmetry at origin
f(-x)=-f(x)
when should you use L’hopital’s ruke?
when the limit of f(x) and g(x) as x approaches c both equal zero OR infinity/negative infinity
what is l’hopital’s rule?
lim of f(x)/g(x) as x approaches c = lim of f’(x)/g’(x) as x approaches c
what should we do for indeterminate products (0 x infinity) or differences (infinity - infinity)?
rewrite into indeterminant quotient and apply L’Hopital
what should we do for indeterminant powers (1^infinty, 0^0, 0^infinity)?
rewrite as e raised to natural log and then indeterminant quotient and apply L’Hopital
