An expression such as
x^3 + 2x^2 - 6x - 2, is a cubic expression.
3x - 4 is linear
x^2 + 5x - 10 is quadratic
These types of expressions can be extended to include higher powers of x, such as
x^5 + 3x^3 - 2x^2 + 3
4x^6 + 2x^4 - x^3 + 5
Expressions of this type are called polynomials.
Note: Polynomials do not have negative or fractional powers. n is a non-negative integer.
of a polynomial is given by the highest power of the variable
An expression of the form
ax^n + bx^(n-1) + ... + px^2 + qx + r
(where a, b, ..., p, q, r are constants is called a polynomial in x)
3x^5 + 3x^4 - 2x^2 -1 is a polynomial of degree 5.
A constant expression is a polynomial of degree 0
A linear expression is a polynomial of degree 1
a quadratic expression is a polynomial of degree 2
a cubic expression is a polynomial of degree 3
, we often arrive at answers that contain root signs (they may be square roots, cube roots, etc.).
We will find that some of these numbers with a root
sign are easy to deal with since they have an exact decimal representation
For instance sqrt
16 = 4, rt^3(8)= = 2, sqrt11.56 = 3.4, rt^5(1/32) = 0.5.
This is because each of these numbers is rational
Expressions with root signs involving pirrational numbers] such as sqrt7 - 2 or rt^3(5) are called surds.
the reverse process of differentiaton
we know that, for example if f(x)
= 2x^3 - 5x^2 + 3x -7
= 6x^2 - 10x + 3
This process can be reversed.
In general, y = x^n -> dy/dx
So, reversing this process, it would seem that dy/dx
= x^m -> y = (1/(m+1))x^(m+1)
The general process of finding a function
from its derivative
is known as interation.
Given that dy/dx = 12x^2 + 4x - 5, find an expression for y.
y = 12((x^3)/3) + 4((x^2)/2) - 5((x^1)/1)
It would seem that
y=4x^3 + 2x^2 - 5x
but that is not quite the complete answer
Whenever you differentiate a constant you get zero,
e.g. y = 7 dy/dx = 0
and so the expression for y above could have any constant on the end and still satisfy dy/dx = 12x^2 + 4x - 5
The answer to this example is therefore
y= 4x^3 + 2x^2 - 5x + c, where c is a constant.