**§9.** By the Pythagorean theorem the **distance** from
to is

The distance from a point P(x, y) to the point
is exactly a when

This is the equation for the circle centered at
with radius a. (See Appendix
C

page A17.) For example, the equation of the circle of radius 5 centered at

is

(x − 2)^{2} + (y − 3)^{2} = 5^{2}, or x^{2} + y^{2} − 4x − 6y − 12 = 0.

The latter equation has the form of a general 2nd degree equation

Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0

(with A = C = 1, B = 0, D = −4, E = −6, F = −12); ellipses (including

circles), parabolas, and hyperbolas all have equations of this form.

**4 Precalculus**

This material will be reviewed as we need it, but you should have

seen it before.

**§10. **A** function** is a rule which produces an output f(x) from an input x. The

set of inputs x for which the function is defined is called the **domain** and f(x)

(read “f of x”) is the **value** of f at x. The set of all possible outputs f(x) as
x

runs over the domain is called the **range** of the function. For example, for the

function f(x) = 1/x^{2} the domain is the set of all nonzero real numbers x (the

value f(0) is not defined because we don’t divide by zero) and the range is the

set of all positive real numbers (the square of any nonzero number is positive).

For the function y = sinθ the domain is the set of all real numbers θ and the

range is the set of all real numbers y such that −1≤ y ≤1.

**§11. **The **graph** of a function

y = f(x)

is the set of all points P(x, y) whose coordinates (x, y) satisfy the equation y
=

f(x). More generally, an equation of form

F(x, y) = 0

determines a set (graph) in the (x, y)-plane consisting of all points P(x, y)
whose

coordinates (x, y) satisfy the equation. The graph of a function y = f(x) is a

special case: take F(x, y) = y−f(x). To decide if a set is the graph of a
function

we apply the

**Vertical Line Test.** A set in the (x, y)-plane is the graph of a

function if and only if every vertical line x = constant intersects

the graph in at most one point. [If the number k is in the domain,

the vertical line x = k intersects the graph y = f(x) in the point

P(k, f(k)).]

**§12**. For example, the graph of the equation x^{2} + y^{2} = 1 is a circle; it is not

the graph of a function since the vertical line x = 0 (the y-axis) intersects
the

graph in two points and
. This graph is however the union of

two different graphs each of which is the graph of a function:

**§13. **In calculus, we learn to reason about a function even when we cannot find

an explicit formula for it. For example, in theory the equation x = y^{5}+y (which

has the form x = g(y)) can be rewritten in the form y = f(x) but there is no

formula for f(x) involving the mathematical operations studied in high school.

**§14.** When two functions are related by the condition

y = f(x) x = g(y)

(for all appropriate x and y) we say that the functions are **inverse** to one

another and write g = f^{ -1}. Then the range of f is the domain of g and vice

versa. Often a function f has an inverse function g only after we modify f

by artificially restricting its domain. The following table lists some common

functions (suitably restricted) and their inverses.

(The notations arcsin y = sin^{-1} y, arccos x = cos^{-1} x, and arctan u = tan^{-1} u

are also commonly used for the inverse trigonometric functions. The exponential

function y = a^{x} is only defined for a positive.) To decide if a function has an

inverse we apply the

**Horizontal Line Test.** A function has an inverse if and only if

every horizontal line y = constant intersects the graph in at most

one point. [The horizontal line y = k intersects the graph y = f(x)

in the point P(f^{ -1}(k), k).]

**5 Trigonometry**

There is a review of trigonometry in section 2-9 of the text but you

should already be familiar with this material. You should also have

seen the inverse trigonometric functions before. These are reviewed

in section 6-2.

**§15.** In calculus we always measure angles in radians rather than degrees. The

radian measure of an angle is the arclength of a circle of radius one (centered

at the vertex of the angle) cut out by the angle. Since the total length of the

circumference of a circle is 2π we get

2π radians = 360°.

(Since 2π = 6.283 . . . is about 6, this means that one
radian is a little less than

60 degrees.)

Let P(x, y) be a point of the (x, y) plane, O(0, 0) denote the origin, r = |OP|,

and θ denote the angle between the positive x-axis and the ray OP. Then

Since r is positive (· always means the positive square root) these formulas

make it easy to remember the symmetries

(which say that the sine is an odd function and the cosine is an even function)

and the sign reversals

All the trigonometric functions have period 2π :

Because of the above sign reversal formulas for the sine and cosine and the

equations

the tangent and cotangent have period π:

The cofunction of an angle is the function of its complement:

.

The trigonometric addition formulas are