Algebraic
Limits
• lim f (x) = L if for every ε > 0 there exists δ > 0 so
• a2 − b2 = (a − b)(a + b)
x→a
that |f (x) − L| < ε when |x − a| < δ.
• a3 − b3 = (a − b)(a2 + ab + b2 )
√
−b ± b2 − 4ac
• Quadratic Formula:
2a
• lim f (x) exists if and only if
x→a
lim f (x) = lim f (x)
x→a−
x→a+
sin θ
=1
θ
1 − cos θ
• lim
=0
θ→0
θ
• lim
Geometric
θ→0
• Area of Circle: πr2
• Circumference of Circle: 2πr
Derivatives
• Circle with center (h, k) and radius r:
f (x + h) − f (x)
h
• (f g)0 = f g + f g 0
0
f 0 g − f g0
f
=
•
g
g2
0
• (f (g(x))) = f 0 (g(x)) · g 0 (x)
• f 0 (x) = lim
(x − h)2 + (y − k)2 = r2
h→0
0
• Distance from (x1 , y1 ) to (x2 , y2 ):
p
(x1 − x2 )2 + (y1 − y2 )2
• Area of Triangle:
• (sin x)0 = cos x
1
2 bh
• (cos x)0 = − sin x
opposite leg
hypotenuse
adjacent leg
• cos θ =
hypotenuse
opposite leg
• tan θ =
adjacent leg
• If 4ABC is similar to 4DEF then
• sin θ =
• (tan x)0 = sec2 x
• (sec x)0 = sec x · tan x
Theorems
• (IVT) If f is continuous on [a, b], f (a) 6= f (b), and N is
between f (a) and f (b) then there exists c ∈ (a, b) that
satisfies f (c) = N .
AB
BC
AC
=
=
DE
EF
DF
• Volume of Sphere:
4
3
3 πr
• Surface Area of Sphere: 4πr2
• Volume of Cylinder/Prism: (height)(area of base)
• Volume of Cone/Pyramid:
1
3 (height)(area
• (MVT) If f is continuous on [a, b] and differentiable
on (a, b) then there exists c ∈ (a, b) that satisfies
f (b) − f (a)
f 0 (c) =
.
b−a
Z x
• (FToC P1) If F (x) =
f (t) dt
of base)
then F 0 (x) = f (x).
Other Formulas
Trigonometric
• sin2 θ + cos2 θ = 1
• sin(2θ) = 2 sin θ cos θ
• cos(2θ) = cos2 θ − sin2 θ
= 1 − 2 sin2 θ
= 2 cos2 θ − 1
a
f (xn )
f 0 (xn )
• Linearization of f at a: L(x) = f (a) + f 0 (a)(x − a)
n
X
•
c = cn
• Newton’s Method:
xn+1 = xn −
i=1
•
n
X
i=1
•
n
X
i=1
i=
n(n + 1)
2
i2 =
n(n + 1)(2n + 1)
6