223
Mathematical Review
Algebra
Quadratic equation:
The solutions of
ax
2
+bx+c=0
are
x=–b
±
b
2
–4ac
2a
.
Logarithms and exponentials:
ln(ab)=lna+lnb
e
a+b
=e
a
e
b
lne
x
=e
lnx
=x
lna
b
=blna
Geometry, area, and volume
area of a triangle of base b and height h=
1
2
bh
circumference of a circle of radius r= 2
p
r
area of a circle of radius r=
p
r
2
surface area of a sphere of radius r=
4p
r
2
volume of a sphere of radius r=
4
3
p
r
3
Trigonometry with a right triangle
.
h = hypotenuse
o = opposite
side
a = adjacent side
Definitions of the sine, cosine, and tangent:
sin
.
=o
h
cos
.
=a
h
tan
.
=o
a
Pythagorean theorem:
h
2
=a
2
+o
2
Trigonometry with any triangle
A
B
C
a
ß
.
Law of Sines:
sin
a
A
=sin
ß
B
=sin
.
C
Law of Cosines:
C
2
=A
2
+B
2
–2ABcos
.
Properties of the derivative and integral
(for students in calculus-based courses)
Let f and g be functions of x, and let c be a constant.
Linearity of the derivative:
d
dx
cf
=cdf
dx
d
dx
f+g
=df
dx
+dg
dx
The chain rule:
d
dx
f(g(x))=f
'
(g(x))g
'
(x)
Derivatives of products and quotients:
d
dx
fg
=df
dx
g+dg
dx
f
d
dx
f
g
=
f
'
g
–
fg
'
g
2
Some derivatives:
d
dx
x
m
=mx
m–1
(except for m=0)
d
dx
sinx=cosx
d
dx
cosx=–sinx
d
dx
e
x
=e
x
d
dx
lnx=1
x
The fundamental theorem of calculus:
df
dx
dx=f
Linearity of the integral:
cf(x)dx
=cf(x)dx
f(x)+g(x)
dx
=f(x)dx
+g(x)dx
Integration by parts:
fdg
=fg–gdf