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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