Basic Derivatives
where u is a function of x, and a is a constant.
( )
d n
x = nx n−1
dx
d
( sin x ) = cos x
dx
d
( cos x ) = − sin x
dx
d
( tan x ) = sec2 x
dx
d
( cot x ) = − csc2 x
dx
d
( sec x ) = sec x tan x
dx
d
( csc x ) = − csc x cot x
dx
d
1 du
( ln u ) =
dx
u dx
d
1
( log a x ) =
dx
x ln a
d u
u du
e =e
dx
dx
d u( x)
du
u x
a
a ( ) ln a ⋅
=
dx
dx
d
1
du
sin −1 u =
2 dx
dx
1− u
( )
( )
(
)
d
−1
cos −1 x =
dx
1 − x2
d
1
du
−1
tan=
u
⋅
dx
1 + u 2 dx
d
−1
cot −1 x =
dx
1 + x2
d
du
1
sec −1 u
=
⋅
2
dx
u u − 1 dx
(
)
(
)
(
)
(
)
d
−1
csc −1 x =
dx
x x2 − 1
(
)
Differentiation Rules
Chain Rule d  f ( g ( x ))  = f '( g ( x )) g ' ( x )


dx
d
dv
du
(=
uv) u
+v
dx
dx
dx
du
d  u  v − u dv
Quotient Rule   = dx 2 dx
dx  v 
v
Product Rule
Some handy INTEGRALS:
∫
tan
=
x dx ln sec x + C
=
− ln cos x + C
∫
sec x dx= ln sec x + tan x + C
AP CALCULUS
Stuff you MUST know Cold
Curve sketching and analysis
y = f(x) must be continuous at each:
dy
critical point:
= 0 or undefined
dx
inc/dec. function f(x): f ’ > 0, f ’ < 0
concavity ≡ inc/dec slope: f “ > 0, f “< 0
local minimum:
dy goes (–,0,+) or (–,DNE,+) or d 2 y >0
dx 2
dx
local maximum:
dy goes (+,0,–) or (+,DNE,–) or d 2 y <0
dx 2
dx
point of inflection:
f ” = 0 or DNE AND concavity changes
* topic only on BC
Approximation Methods for Integration
Use Geometry formulas
Rectangles - Left, Right and Middle
Riemann Sums A = bh
Trapezoids: A = ½ (b1 + b2)h
(Effects of inc/dec & cocavity on approx.)
Concave up: M under estimate, T over estimate
Concave down: M over estimate, T under estimate
Inc: L=under, R=over. Dec: L = over, R = under
First Fundamental Th. of Calculus
∫
b
a
)dx f (b) − f (a )
f '( x=
2nd Fundamental Th. of Calculus
d b( x) f (t )dt =
dx ∫a ( x)
d 2 y goes from (+ to –), (– to +),
dx 2
Abs. Max/Min: eval. crit # & endpts.
f (b( x)) b '( x) − f (a ( x)) a '( x)
OR discuss “always inc or always dec.”
Intermediate Value Theorem:
If the function f(x) is continuous on [a, b],
Solids of Revolution and friends
for all k between f(a) and f(b), there exists at Disk Method
b
2
least one number x= c in the open interval
V = π ∫ [ R ( x) ] dx
a
(a, b) such that f (c) = k .
Washer Method
Extreme Value Theorem:
b
2
2
If the function f(x) is continuous on [a,=
b],
V π ∫ [ R( x) ] − [ r ( x) ] dx
a
then there exists an absolute max and min
volume by cross section (not rotated)
on that interval.
b
Rolle’s Theorem:
V = ∫ Area ( x) dx
a
If the function f(x) is continuous on [a, b],
s2
AND differential on the interval (a, b),
3
Aeq . lat . ∆ =
AND f(a) = f(b), then there is at least one
4
b
2
number x = c in (a, b) such that f '(c) = 0
*Arc Length
=
s ∫ 1 + [ f '( x) ] dx
a
Mean Value Theorem: m secant = m tangent
*Surface Area
If the function f(x) is continuous on [a, b],
b
2
then
AND differential on the interval (a, b),
=
SA 2π ∫ radius 1 + [ f '( x) ] dx
a
there is at least one number x = c in (a, b)
such that f '(c) = f (b) − f (a ) .
Distance, Velocity, and Acceleration
b−a
MVT of Integrals i.e. AVERAGE VALUE: velocity = d (position)
If the function f(x) is continuous on [a, b]
dt
and differential on the interval (a, b),
acceleration = d (velocity)
then there exists at least one number
dt
x = c on (a, b) such that
dx dy
b
*velocity vector =
,
f (c ) ( b − a ) =
∫a f ( x)dx
dt dt
Area rectangle = Net Area Integral
speed =
= v
( x ') 2 + ( y ') 2 *
b
1
f (c ) =
f ( x ) dx .
(
(b − a )
)
∫
a
This value f(c) is the “average value”
of the function on the interval [a, b].
Limit Strategies: Factor and cancel,
Rationalize Numerator,
a x u-sub,a HA rules:
=
lim
x →∞
b
bx 2 + c
2 HA
−a
ax
=
lim
2
x →−∞
b
bx + c
To find all HA: Take limit as x → both ± ∞
displacement =
tf
∫t
v dt
o
total distance = ∫
final time
initial time
tf
∫t
v dt
( x ')2 + ( y ')2 dt *
o
Av. velocity =
1 b
b − a ∫a
1 b
v(t )dt = s(b) − s(a)
b − a ∫a
b−a
[ x '(t )] + [ y '(t )]
2
2
dt *
BC TOPICS and important TRIG identities and values:
l’Hôpital’s Rule
Slope of a Parametric equation
Given x(t) and y(t), the slope is
dy
dy dy
d 2 y Dt dt
dt
,
= dx=
dx
dx dt dx 2
dt
f (a) 0
∞
If = =
,
or
g (b) 0
∞
f ( x)
f '( x)
then lim
= lim
x →a g ( x)
x → a g '( x )
( )
Euler’s Method
If given that
dy
dx
Polar Curve
= f ( x, y )
For a polar curve r(θ), the
and the
AREA inside a “leaf” =
solution passes through (xo, yo),
xnew
= xold + ∆x
2
The SLOPE of r(θ) at a given θ is
dy
dy
=
dx
d
dθ
dθ
=
d
dθ
dx
dθ
Integration by Parts
(u=ILATE=dv)
uv − ∫ vdu
f ( x=
) g '( x)dx f ( x) g ( x) − ∫ g ( x) f '( x)dx
Ratio Test:
converges if lim an +1 < 1
n →∞
n →∞
an
If the limit equals 1, you know nothing
so check the endpoints using another test.
converges if lim n an < 1
n →∞
p–series Test:
ln x dx
= x ln x − x + C
≠0
diverges if lim an
Root Test:
Use integration by parts and let u = ln x
 r (θ ) sin θ 
 r (θ ) cos θ 
nth Term:
Integral of Ln
∫
 r (θ )  dθ
θ1 and θ2 are the “first” two times that r = 0.
 dy

 (∆x) + yold
=
ynew 
 dx ( x , y ) 
old old 

∫
θ
dr
determines inc/dec and relative max/mins.
dθ
Basically Algebra I:
y = m(x – x1) + y1
∫ udv=
∫
2
1
2 θ
1
∞
∑
n =1
1 converges if p > 1
pn
Alternating: converges if alt. & lim an = 0
Values of Trigonometric
Functions for Common Angles
θ
sin θ
cos θ
tan θ
0
0
0°
1
1
π
3
3
6
2
2
3
π
2
2
1
4
2
2
π
1
3
3
3
2
2
π
2
π
1
0
“∞ ”
0
−1
0
Know both the inverse trig and the trig values.
π
Careful: tan  3π  =
−
  −1 but arctan ( −1) =
4
 4 
Trig Identities
Double Angle
sin 2 x = 2sin x cos x
cos 2 x =
cos 2 x − sin 2 x =
1 − 2sin 2 x
Power Reduction
(1 − cos 2 x )
sin 2 x =
2
(1 + cos 2 x )
cos 2 x =
2
n →∞
∞
Taylor Series
Geometric Series:
If the function f is “smooth” (continuous
and differentiable) at x = c,
then it can be approximated
by the nth degree polynomial
∑a r
n=0
n
1
converges if |r|<1
Limit Comparison: If lim an exists and ≠ 0,
n →∞
bn
what one series does (C or D), so does the other.
Direct Comparison: If 0 < an < bn and
Pythagorean
sin 2 x + cos 2 x =
1
(others are easily derivable by
dividing by sin2x or cos2x)
1 + tan 2 x =
sec 2 x
cot 2 x + 1 =
csc 2 x
∞
Reciprocal
if ∑ a diverges, then ∑ bn diverges;
f ''(c)
f ( n ) (c )
2
n
+
( x − c) +  +
( x − c) .
1
n =1
=
sec x =
or cos x sec x 1
2!
n!
∞
∞
cos x
if ∑ bn converges then ∑ a converges.
n
1
n =1
n =1
=
csc x =
or sin x csc x 1
sin
x
Maclaurin Series (Taylor Series about x = 0)
Error Bound (Remainder)
Odd-Even
Alternating Series:
x 2 x3
xn
e x =1 + x +
+ + +
+ ...
N
sin(–x) = – sin x (odd)
n
2! 3!
n!
If S=
−1) an is the Nth partial sum of a
(
∑
N
cos(–x) = cos x
(even)
n
k =1
x 2 n ( −1)
x2 x4
tan(–x) = –tan x (odd)
cos x =1 − +
− +
+ ...
convergent alternating series,
2! 4!
( 2n )!
RN ≤ aN +1 (the next term)
Infinite Sums
n
2 n +1
3
5
x ( −1)
x
x
Lagrange Error Bound of a Taylor Series:
sin x =x − + −  +
+ ...
Geometric Series: S∞ = a1 if |r|<1
Let c = # centered on and x = value you want to
3! 5!
( 2n + 1)!
1− r
approximate. There exists a z between x and c,
Telescoping
Series:
Expand
& cancel
1
f n +1 ( z )
∞
=1 + x + x 2 + x 3 +  + x n + ...
n +1
1
such that Rn ≤
x−c
1− x
Special Series:
= e1
( n + 1)!
1
2
3
n
n +1
f ( x) ≈ f (c) + f '(c)( x − c) +
∞
n
n =1
ln(=
x)
( x − 1) ( x − 1) ( x − 1)
1
−
2
+
3
− +
( x − 1) ( −1)
n
+ ...
Find interval of
f
n +1
( z ) to find error interval.
∑ n!
n =0