# Math Review

## 1 Geometry

• Here are some areas of standard geometric figures that you should remember:

• The volume of a sphere is (4/3)πR^{3}, its surface area is 4πR^{2}.

• The circumference of a circle is 2πR, where R is the radius, or πD, where D
is the diameter

(D = 2R).

• The volume of a right figure is is V = A × h, where h is the height of the
figure, and A is

the area of the base. If the base is a circle, then we’re talking about a right
cylinder and

A = πR^{2}. If the base is a rectangle, you get the idea.

## 2 Algebra

• Solving linear equations is easy: Ax + B = C implies the solution for x is
x = (C − B)/A.

Solving quadratic equations: Ax^{2} + Bx + C = 0 implies

• Solving a system of 2 equations with 2 unknowns: if

(where the A_{i}'s, etc., are constants) then you can “solve” the first equation
for x like x =

(C_{1} − B_{1}y)/A_{1} and insert this expression into the second equation. Then you will
have a

solution for y. Once you know y, you can use this value for y to solve for x.

## 3 Trigonometry

• In the figure, the trigonometric functions sine, cosine, and tangent acting
on the angle θ are

defined as follows

• A consequence of Pythagoras’ theorem is that sin^{2}θ + cos^{2}θ
= 1

• These are some useful values of the trig functions you should remember I
reminded you of,

but I will generally put these on exams for you:

• Radian vs. Degree Measure of Angles: In the figure above, if c = 1, then
the circle drawn is

called the unit circle. The relationship between the length s of a piece of arc
and the radius

R and angle of arc θ is s = Rθ. When this relation is applied to the unit
circle, we see that

the angle θ=1 revolution has to be measured in a very particular way: in
“radians”. One

revolution = 360◦ = 2π “radians”.

1. 0◦ = 0 radians

2. 45◦ = π/4 radians

3. 90◦ = π/2 radians

4. 180◦ = π radians

5. 270◦ = 3π/2 radians

6. 360◦ = 2π radians

• You should convince yourself that you understand the graphs below. I don’t
expect you to to

memorize them, but they should make sense if you look at the particular values
of the sines

and cosines listed above (when degrees are converted to radians).

## 4 Calculus

• A function maps each possible value of an independent variable t to a value
of the dependent

variable x. We write x = f(t), or equivalently x = x(t).

• The derivative of the function x(t) at time t is defined by

A common notation for “the derivative of x(t) with respect to t” is dx/dt .
The derivative dx/dt

gives the slope of x(t). For example, the derivative of the function f(x) shown
in the figure

below is zero at x = 5 and x = 18. The derivative is positive between these two
points and

negative outside these two points.

• An antiderivative of f(t) is a function F(t) such that F'(t) = dF/dt = f(t).
Any antiderivative

plus a constant C is also an antiderivative, since the derivative of a constant
is zero. A

common notation for all the antiderivatives of f(t) is
,
which is called the indefinite

integral.

• The definite integral of a function f(t) from t_{1} to t_{2} is

where F(t) is an antiderivative of f(t). The definite integral gives the area
under the function

between the limits of integration (t_{1} and t_{2}).

• For a function of the form x(t) = at^{n}:

• For some common functions: