Studied by 1 personln MN
ln M + ln N
ln (M/N)
ln M - ln N
ln b^a
a ln b
e^ln x
x
sin^2(x) + cos^2(x) =
1
lim (sin ax)/bx
x→0
a/b
lim (sin x)/x
x→inf
0
lim (cos x - 1)/x
x→0
0
dy/dx
(dy/dt)/(dx/dt)
d/dx (f(g(x))
f'(g(x)) * g'(x)
d/dx (c)
0
d/dx (u^n)
n u^(n-1) * u'
d/dx (x)
1
d/dx (x^n)
n x^(n-1)
d/dx (cx)
c
d/dx (u * v)
(u' * v) + (v' * u)
d/dx (u/v)
(vu' - uv')/v^2
position
s(t)
velocity
v(t) = s'(t)
acceleration
a(t) = v'(t) = s''(t)
jerk
a'(t) = s'''(t)
limit definition of derivative
lim (f(x + h) - f(x))/h
h→0
alternate limit definition of derivative
f'(x)=lim (f(x) - f(a))/x-a
x→a
intermediate value theorem
if function f is continuous through every x-value in [a,b], then it takes on every y-value in [f(a),f(b)]
continuity at a point
differentiability
d/dx sin u
u' cos u
d/dx cos u
-u' sin u
d/dx tan u
u' sec^2 u
d/dx sec u
u' sec u tan u
d/dx csc u
-u' csc u cot u
d/dx cot u
-u' csc^2 u
sin (0)
0
cos (0)
1
tan (0)
0
sin (π/6)
1/2
cos (π/6)
√3/2
tan (π/6)
√3/3 or 1/√3
sin (π/4)
1/√2
cos (π/4)
1/√2
tan (π/4)
1
sin (π/3)
√3/2
cos (π/3)
1/2
tan (π/3)
√3
sin (π/2)
1
cos (π/2)
1
tan (π/2)
undefined
sin (π)
0
cos (π)
-1
tan (π)
0
sin (2π)
0
cos (2π)
1
tan (2π)
0
sin (3π/2)
-1
cos (3π/2)
0
tan (3π/2)
undefined
sec (0)
1
csc (0)
undefined
cot (0)
undefined
sec (π/6)
2/√3 or 2√3/3
csc (π/6)
2
cot (π/6)
√3
sec (π/4)
√2
csc (π/4)
√2
cot (π/4)
1
sec (π/3)
2
csc (π/3)
2√3/3
cot (π/3)
√3/3
sec (π/2)
undefined
csc (π/2)
1
cot (π/2)
1
sec (π)
-1
csc (π)
undefined
cot (π)
undefined
sec (3π/2)
undefined
csc (3π/2)
-1
cot (3π/2)
0
sec (2π)
1
csc (2π)
undefined
cot (2π)
undefined
d/dx arcsin u
1/√(1-u^2) * u'
d/dx arccos u
1/√(1-u^2) * u'
d/dx arctan u
1/(u^2+1) * u'
d/dx arccot u
-1/(u^2+1) * u'
d/dx arcsec u
1/(|u|√(u²-1)) * u' ; |u|>1
d/dx arccsc u
-1 /(|u|√(u²-1)) * u' ; |u|>1
df^-1/dx |x=f(a)
1/(df/dx) |x=a
d/dx e^u
u' * e^u
d/dx b^u
u' * b^u * ln b
d/dx ln u
(1/u)u'
d/dx logb (u)
(1/(u ln b) ) * u'
l"Hospital's Rule (for when the limit is indeterminate)
lim x->a (f(x)/g(x)) = lim x->a (f'(x)/g'(x))
average rate of change (over interval [a,b])
(f(b)-f(a))/b-a
instantaneous rate of change
if (a, f(a)) is a point on the graph of y=f(x), then the instantaneous rate of change of y with respect to x is f'(a)
extreme value theorem
If f is continuous over a closed interval, then f has a maximum and minimum value over the interval
(mean value theorem for derivatives) If f(x) is continuous in [a,b] and differentiable in (a,b) then...
at some point c in (a,b), f'(c) = (f(b)-f(a))/b-a
(first derivative test) If f'(x) > 0 for every x in (a,b) then...
f(x) is increasing on [a,b]
(first derivative test) If f'(x) < 0 for every x in (a,b) then...
f(x) is decreasing on [a,b]
(first derivative test) If f'(c) changes sign through x = c then...
f has a local maximum/minimum at x=c
(second derivative extremum test) If f'(c) = 0 and f''(c) > 0 then...
f(c) is a local minimum
Note
Flashcard