AP Review Flashcards
(20 cards)
definition of a derivative
f’(x)=lim h-0 f(x+h)-f(x)/h
definition of continuity
f is continuous at c if :
1. f(c) is defined
2. lim x-c f(x) exists
3. lim x-c f(c)=f(c)
MVT
if f is continuous on [a,b] and differentiable on (a,b), there exists a number c on (a,b) such that f’(c)=f(b)-f(a)/b-a
IVT
if f is continuous on an interval and k is between a and b then there is at least one number c between a and b such that f(c)=k
trig/special derivatives
sin
cos
tan
cot
sec
csc
ln u
loga u
a^u
arcsin
arccos
arctan
arccot
cos
-sin
sec^2
-csc^2
sectan
-csccot
1/u
1/u lna
a^u lna
1/ root 1-u^2
-1/root 1-u^2
1/1+u^2
-1/1+u^2
trig/special integrals
cos
sin
x^n
1/u
a^u
sin
-cos
x^n+1/n+1
ln|u|
a^u/lna
critical number
when f’(c)=0 or undefined at c then c is a critical number of f
first derivative test
if f’(x) changes from + to - at c then c is a relative max
if f’(x) changed from - to + at c then c is a relative min
second derivative test
- if f’(x)=0 and f”(x)>0 then c is a relative min
- if f’(x)=0 and f”(x)<0 then c is a relative max
definition of concavity
Let f be differentiable on an open interval I. The graph of f is concave upward on I if
f is increasing on the interval and concave downward on I if f is decreasing on the interval.
test for concavity
1) If
f “x > 0 for all x in I, then the graph of f is concave upward in I.
2) If f “x <0 for all x in I, then the graph of f is concave downward in I.
definition of an inflection point
- if f”(c)=0 or does not exist and
- if f” changes sign from + to - or - to + at x=c OR if f’(x) changes from increasing to decreasing or decreasing to increasing at x=c
average rate of change of f(x)
f(b)-f(a)/b-a
average VALUE of f(x)
1/b-a integral from a to b f(x)
particle motion
v(t)=s’(t)
speed= |v(t)|
a(t)=v’(t)=s”(t)
displacement: ∫ v(t) dt
total distance: ∫|v(t)|
-an object is speeding up when velocity and acceleration have the same sign
-an object is slowing down when velocity and acceleration have opposite signs
integration by parts
∫udv=uv-∫vdu
area under a curve
∫[upper-lower]
volume
disk method:
V=pi∫(upper-lower)^2
washer method:
V= pi ∫[R^2 - r^2]
shell method:
V= 2pi ∫r(x)h(x)
length of a curve
L= ∫ root 1+(dy/dx)^2
or
L= ∫ root 1+ (dx/dy)^2
definition of taylor polynomial
f^n(c) (x-c)^n / n!
