Polar Cartesian Equations
Slope in Polar
d tan x dx
d sec x dx
d csc x dx
d cot x dx
d sin 1 x dx
d cos 1 x dx
d tan 1 x dx
d sec1 x dx
d csc1 x dx
d cot 1 x dx
sin x x x
d x 3 dx
Indeterminate forms (L’Hopital’s Rule)
sin x x 0 x
lim
lim
sec 2 x
sec x tan x
1 1 x2
1 1 x2
dy d after x r cos dx y r sin d
1 1 x2
1 x x 1 2
1 x x 1 2
x2 y 2 r 2
csc x cot x
csc 2 x
x r cos y r sin
1 1 x2
3x ln 3
0
1
0 , , 0 , , 0
0 , 0 , 1 0
n
1 lim 1 n n
Euler’s Method
Newton’s Method
u dv
Mean Value Theorem
Average Value
Arc Length (Cartesian)
Arc Length (Parametric)
Area (Polar)
Slope in Parametric And Concavity in Parametric
Surface Area (Cartesian)
Logistic Growth
Exponential Growth
Growth Order as x
Speed (Parametric)
d log 3 x dx
Integration by parts
To estimate x‐intercept
f xn xn 1 xn f ' xn
To estimate a point when given only the slope field
1 x ln 3
e
2
dy 1 dx dx
b
a
f ( x) dx
Or
a
c
dx 1 dy dy
uv v du
f (b) f (a) f '(c) ba
ba
2
d
1. If f is continuous on [a, b] 2. If f is differentiable on (a, b) then there is a c such that
b
2
dy 2 r 1 dx dx a b
d dy dy dt dx Slope: dt Conc: dx dx dt dt
Or 2
dx 2 r 1 dy dy c d
b
1 2 r d 2 a
b
a
2
2
dy dx dt dt dt
when r = radius of revolution 2
2
dy dx dt dt Log Polynomial Exponents Factorial n n ln n ln ln n
n2 1
en
n
3
n! n
dP kP dt
dP P kP 1 dt M
Divergence Test
Alternating Series Remainder
Geometric Series
Alternating Series Test
Integral Test
Comparison Test
Limit Comparison Test
Ratio Test
Taylor Series
Maclaurin Series
Maclaurin Series for ex
Maclaurin series for sin x
Maclaurin series for cos x
Root Test
P‐Series
LaGrange Error Bound
If the alternating series
1
n 1
n 1
bn
satisfies (i) bn 1 bn (deriv. neg) (ii) lim bn 0 ,
A series in the form,
an 1 L 1 , then the series an
is absolutely convergent (ii) If lim n
an 1 L 1 or , then the an series is divergent
a (iii) If lim n 1 1 , inconclusive n a n
x
x3 x5 x 7 ... 3! 5! 7!
x 2 n 1 1 2n 1! n 0 n
If s
s
1
n 1
If lim an 0 or DNE, then the series n
bn and
a
(i) bn 1 bn and (ii) lim bn 0 ,
If r 1 , the series converges to
n
n 1
Then Rn s sn bn 1
first term 1 r
f ( n 1) (c) n 1 Rn ( x) x a n 1! For some c between a and x
is divergent.
(If lim an =0, inconclusive) n
Suppose that
a
n
(given) and
b
n
(selected) are series with positive terms
a If lim n c 0 , then either both n b n series converge or both series diverge
Suppose that
a
n
x x 2 x3 ... 1! 2! 3!
b
n
(selected) are series with positive terms (i) If an bn and bn is conv, then
an is conv (ii) If an bn and
a
f (0)
bn is div, then
is div
n
1
(given) and
f '(0) f ''(0) 2 f (3) (0) 3 x x x ... 1! 2! 3!
xn n 0 n !
A series in the form,
Suppose f is continuous, positive, decreasing on [1, ) and an f (n) , then
If
f ( x)dx is conv, then an is conv n 1
1
If
f ( x)dx is div, then an is div n 1
1
f (a)
f '(a) f ''(a ) f (3) (a) 2 3 x a x a x a ... 1! 2! 3!
n 0
f ( n ) (0) n x n!
(i) If lim n an L 1 , then the series In a Taylor Series
n
.
If r 1 , the series diverges
then the series converges. (If not, try Divergence Test)
n
n 1
n 1
n
(i) If lim
a r
1
n
p
n
If p 1 , the series diverges If p 1 , the series converges
is absolutely convergent (ii) If lim n an L 1 or , then the n
series is divergent (iii) lim n an 1 , inconclusive n
=
f
n 0
1
(n)
(a) n x a n!
x2 x4 x6 ... 2! 4! 6!
1 n0
n
x2n 2n !
