434 Formulas and Theorems
- Double Angles:
- sin 2θ=2 sinθcosθ
- cos 2θ=cos^2 θ−sin^2 θor
1 −2 sin^2 θor 2 cos^2 θ− 1. - cos^2 θ=^1 +cos 2θ
2 - sin^2 θ=^1 −cos 2θ
2
- Pythagorean Identities:
- sin^2 θ+cos^2 θ= 1
- 1 +tan^2 θ=sec^2 θ
- 1 +cot^2 θ=csc^2 θ
- Limits:
xlim→∞
1
x
= 0 limx→ 0
cosx− 1
x
= 0
limx→ 0
sinx
x
= (^1) hlim→∞
(
1 +
1
h
)h
=e
limh→ 0
eh− 1
h
= 1 limx→ 0 ( 1 +x)
(^1) x
=e
- L’Hoˆpital’sRule for Indeterminate Forms
Letlimrepresent one of the limits:
limx→c, limx→c+, limx→c−, limx→∞,orxlim→−∞. Suppose
f(x) andg(x) are differentiable, andg′(x)/= 0
nearc, except possibly at c, and suppose
limf(x)=0 and limg(x)=0, then the
lim
f(x)
g(x)
is an indeterminate form of the type
0
0
. Also, if limf(x)=±∞and
limg(x)=±∞, then the lim
f(x)
g(x)
is an
indeterminate form of the type
∞
∞
. In both
cases,
0
0
and
∞
∞
,L’Hoˆpital’sRule states that
lim
f(x)
g(x)
=lim
f′(x)
g′(x)
.
- Rules of Differentiation:
a. Definition of the Derivative of a Function:
f′(x)=limh→ 0
f(x+h)− f(x)
h
b. Power Rule:
d
dx
(xn)=nxn−^1
c. Sum & Difference Rules:
d
dx
(u±v)=du
dx
±
dv
dx
d. Product Rule:
d
dx
(uv)=v
du
dx
+u
dv
dx
e. Quotient Rule:
d
dx
(u
v
)
=
v
du
dx
−u
dv
dx
v^2
, v=/ 0
Summary of Sum, Difference, Product,
and Quotient Rules:
(u±v)′=u′±v′ (uv)′=u′v+v′u
(u
v
)′
=
u′v−v′u
v^2
f. Chain Rule:
d
dx
[f(g(x))]= f′(g(x))·g′(x)
or
dy
dx
=
dy
du
·
du
dx
- Inverse Function and Derivatives:
(
f−^1
)′
(x)=
1
f′(f−^1 (x))
or
dy
dx
=
1
dx/dy
- Differentiation and Integration Formulas:
Integration Rules:
a.
∫
f(x)dx=F(x)+C⇒F′(x)=f(x)
b.
∫
af(x)dx=a
∫
f(x)dx
c.
∫
−f(x)dx=−
∫
f(x)dx