Riemann sums A Riemann sum R(f, P, C) for the interval [a, b] is defined by choosing
a partition P:a = x0 < x1 < x2 < < xN = b, and sample point C = cfw_ci, where ci in [xi-1, xi].
Let delta xi = xi xi-1, then R(f, P, C) = f(ci)xi.
Partition The maximum o
Mean Value Theorem (MVT) if f(x) is continuous in an interval [a, b], then there must be a line
tangent to point c in [a, b] parallel to line ab. The conclusion also can written as formula
f(b)- f(a) = f(c) (b-a)
For example, for function f(x) = x2 contin
linear approximation linear approximation uses the derivative to estimate the
without computing it exactly. When the delta x is small and close to 0, the
value derivative of f(a) will be the limit at that point.
. So base on the
condition, f is differenti
transition points a point in the domain of f at which either f changes
sign(local min or max) or f changes sign(point of inflection).
For example, in the graph above, those point are transition point which mark
the change of graph. The red point represe
Antiderivatives Given the derivative, find the function itself is antiderivative. The official
definition is that a function F(x) is an antiderivative of f(x) on (a, b) if F(x) = f(x) for all x
belongs to (a, b).
For example: F(x) = - cos(x) is an antider
Implicit differentiation If y is determined by an equation related to x and y, we say that y is
defined implicitly. And the method we use to solve the derivative of y is called implicit
Here is the steps for implicit differentiation:
second derivative - in a word, second derivative is to find the derivative of the
derivative function from the original function. It usually denoted by y' or f'(x). It is
physical meannig is the rate of change of f'(x)
For example: if the original functio
difference quotient - expression is
. It resprent the slope of lie throught
points (a, f(a), and points (b, f(b). As the graph shown below, the slope of line PQ is f
derivative - The derivative of one points a is the limit x->a of its difference quot
derivative of inverse function - A formula to differentiate logarithmic function. if
g(x) = f-1(x), b belongs to the domian of g(x) and f'(g(b) !=0, then g'(b) = 1/f'(g(b).
we can prove that by chain rule. f'(g(x) = x' , f'(g(x)g'(x) = 1 , g'(x) = 1/f'(g(
Product Rule - if a function is the product of two differentiable function g(x) and f(x),
then the dervative of this function will be result of f(x) * g'(x) + g(x) * f '(x).
For example: calculate the dervative of L(x) = 2x(3-2x)
From the L(x), we can see
it is a limit theorem for us to find the limit for speical function.
the condiction is f(x) should stay between two other function and the those two function
have the same limilt at x->c. Then, we can conclude that the lmit for f(x) at x->
[QUOTE=SMOG;13118746]Rumble is red. That is the correct Rumble. Cartoon Rumble is some aberration, a crime against nature, a false idol, a smear on the canon of Cybertronian rectitude.[/QUOTE]
[QUOTE=HARD RAINBOW;12350219]I need a reason to stay mad at Ha
The Boston Massacre: It was a street fight that occurred on March 5, 1770,
between a "patriot" mob, throwing snowballs, stones, and sticks, and a squad of
British soldiers. Several colonists were killed and this led to a campaign by
speech-writers to rous
Im dedicating this poem to you, Mia, because of your
admiration to the trees, you are always talking about
their vital role and why we need to protect them. I
remember that time when you started discussing on this
topic with our cousins, and how you criti