Math 185
Calculus II
Formulas
If u and v are functions that depend on x we have.
Basic Derivatives
1.
d
d n
(u ) = nun−1 · (u)
dx
dx
− − − − − − − − −−
(Chain Rule)
d
d
d
(u · v) =
(u) · v + (v) · u
− − − − − − − − −−
(Product Rule)
dx
dx
dx
d
d
(u) · v − dx
(v) · u
d u
= dx
3.
− − − − − − − − −−
(Quotient Rule)
2
dx v
v
2.
d
(u)
d
(ln (u)) = dx
4.
dx
u
5.
d
d
(sin u) = cos u · (u)
dx
dx
6.
d
d
(cos u) = − sin u · (u)
dx
dx
7.
d
d
(tan u) = sec2 u · (u)
dx
dx
8.
d
d
(cot u) = − csc2 u · (u)
dx
dx
9.
d
d
(sec u) = sec u · tan u · (u)
dx
dx
10.
d
d
(csc u) = − csc u · cot u · (u)
dx
dx
11.
d
1
d
(sin−1 u) = √
(u)
dx
1 − u2 dx
12.
d
−1
d
(cos−1 u) = √
(u)
dx
1 − u2 dx
13.
1
d
d
(tan−1 u) =
(u)
2
dx
1 + u dx
14.
d
−1 d
(cot−1 u) =
(u)
dx
1 + u2 dx
c 2015
J.M. Villalobos Basic Integrals
un+1
+ C n 6= −1
n+1
Z
Z
2.
udv = uv − vdu + C
− − − − − − − − −−
Z
1.
un du =
Z
du
= ln |u| + C
u
Z
au
a du =
+C
ln a
3.
4.
u
(Integration by parts.)
Z
− − − − − − − − −−
(if a = e then
eu du = eu + C)
Z
sin udu = − cos u + C
5.
Z
6.
cos udu = sin u + C
Z
7.
sec2 udu = tan u + C
Z
2
Trapezoid Rule
b
Z
f (x) ≈
csc udu = − cot u+C
8.
a
h
f (x0 ) + 2f (x1 ) + ... + 2f (xn−1) + f (xn )
2
Z
9.
sec u tan udu = sec u + C
Where h =
b−a
n
Z
csc u cot udu = − csc u + C
10.
Z
tan udu = ln | sec u| + C
11.
Z
Simpson’s Rule
Z
cot udu = ln | sin u| + C
12.
b
f (x) ≈
a
(n
is even)
h
[f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 ) + ...
3
Z
sec udu = ln | sec u + tan u| + C
13.
Z
csc udu = ln | csc u − cot u| + C
14.
Z
15.
a2
du
1
u
= tan−1 + C
2
+u
a
a
c 2015
J.M. Villalobos + 2f (xn−2 ) + 4f (xn−1) + f (xn )]
Volume
Z
b
1.
π[f (x)2 − g(x)2 ]dx
−−−−−−
Disk/Washer Method about the x−axis
π[p(y)2 − q(y)2 ]dy
−−−−−−
Disk/Washer Method about the y−axis
a
Z
b
2.
a
Z
b
3.
2πx(f (x) − g(x))dx
−−−−−−
Shell Method about the y−axis
2πy(p(y) − p(y))dy
−−−−−−
Shell Method about the x−axis
a
Z
4.
b
a
Trigonometry Identities
2
Arc Length for y = f (x)
2
1. sin x + cos x = 1
L=
Z bp
a
2. 1 + tan2 x = sec2 x,
sin 2x = 2 sin x cos x
3. 1 + cot2 x = csc2 x
4. sin2 x =
1 − cos (2x)
2
5. cos2 x =
1 + cos (2x)
2
Trigonometry Values
−− sin x cos x
0
0
√1
π
1
3
6
√2
√2
π
2
2
4
2
√2
π
3
1
3
2
2
π
1
0
2
π
0
−1
3π
−1
0
2
2π
0
1
c 2015
J.M. Villalobos tan x csc x sec x cot x
und
1
und
√0
√
3
2
√
2
3
3
3
√
√
1
2
2
1
√
√
2
3
√
3
2
3
3
und
1
und
0
0
und
−1
und
und
−1
und
0
0
und
1
und
1 + f 0 (x)2 dx
Sequences and Series
Conclusion
Test
Convergent
ak+1
=L
lim
k→∞ ak
√
lim k ak = L
1. Ratio test
2. Root test
L<1
k→∞
∞
X
1
kp
k=1
3. p-series
p>1
p≤1
∞
L = 1 Inconclusive
Z
∞
f (x)dx = ∞
f (x)dx < ∞
1
1
ak > 0, 0 < ak+1 ≤ ak
5. Alt. Series T.
L = 1 Inconclusive
L>1
ak = f (k)
6. Divergence Test
Comments
L>1
L<1
Z
4. Integral test
Divergent
lim ak = 0
k→∞
Does not apply
lim ak 6= 0
k→∞
lim ak 6= 0
k→∞
Power Series
1. f (x) =
∞
X
f (k) (a)
k!
k=0
x
2. e =
(x − a)k
−−−−−−
Taylor Series
∞
X
xk
k=0
k!
∞
X
1
xk ,
3.
=
1 − x k=0
4. sin x =
∞
X
(−1)k x2k+1
k=0
5. cos x =
(2k + 1)!
∞
X
(−1)k x2k
k=0
−1
6. tan
|x| < 1
x=
(2k)!
∞
X
(−1)k x2k+1
k=0
7. − ln (1 − x) =
2k + 1
∞
X
xk
k=1
k
,
,
|x| ≤ 1
−1 ≤ x < 1
Parametric and Polar Curves
0
dy
f (θ0 ) sin θ0 + f (θ0 ) cos θ0
1.
= 0
dx
f (θ0 ) cos θ0 − f (θ0 ) sin θ0
- dy = 0 −→ horizontal tangent line.
- dx = 0 −→ vertical tangent line.
Z β
1
2.
(f (θ)2 − g(θ)2 )dθ
−−−−−−
2
α
c 2015
J.M. Villalobos Area of regions in polar coordinates.