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
=elimh→ 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 anindeterminate form of the type∞
∞
. In both
cases,0
0
and∞
∞
,L’Hoˆpital’sRule states thatlim
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)
hb. 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
dxe. Quotient Rule:d
dx(u
v)
=v
du
dx
−u
dv
dx
v^2, v=/ 0Summary 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)dxc.∫
−f(x)dx=−∫
f(x)dx