Area Under a Curve

What does "area under a curve" tells us?
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The area under the plasma (serum, or blood) concentration versus time curve (AUC) has an number of important uses in toxicology, biopharmaceutics and pharmacokinetics. Toxicology AUC can be used as a measure of drug exposure. It is derived from drug concentration and time so it gives a measure how much - how long a drug stays in a body.
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The work (W) done by a constant force (F) acting on a body by moving it through a distance (d) is given by: W = F × d
1) When the work is done by a CONSTANT FORCE AS IN: An ball about 1 N. If you lift the ball 1 m above a table, you have done approximately 1 Newton metre (Nm) of work.
2) Sometimes WORK is done by a Variable Force If the force varies (e.g. compressing a spring) we need to use calculus to find the work done. If the force is given by F(x) (a function of x) then the work done by the force along the x-axis from a to b is found by using INTEGRATION.
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Integrals with (Math Expressions) ^ POWER

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Finding the Constant of Integration in Calculus

Suppose we have a function y = f(x). It has a derivative of "2". Thus, by integrating "2" we get 2x +C. The orginal function f(x) is represented by y = 2x + C . This is a linear function with constant slope of 2 and Y-INTECEPT of C. Allowing C to take on each REAL NUMBER would create an infinite number of lines that completely cover the xy-plane. If you happen to know that the orginal function y = f(x) passes thru a certain point say (3,10) then "C" has unique value of 4. " y = 2 x + 4" is the only line with slope 2 that will pass thru the point (3,10)
In other words, the INITIAL CONDITIONS OF (3,10) makes the equation "y = 2 x + 4""TRUE"!
Thus, to find C we must have an INITIAL (CONDITION) POINT that f(x) passes through. Plugging the point into the equation with y and x and C will enable us to find the unique value for C that will allow the point to be on the graph of f(x).

HOW ABOUT ANOTHER EXAMPLE:







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Using f and f' and f"

The questions below all refer to a given function … let’s call it f(x)
1) Find the Equation of the SECANT LINE
that passes thru the two points
where x = 0 and x = 2 on the curve f(x).
2) Find the Equation of the TANGENT LINE
that passes thru the point
where x = 2 on the curve f(x).
3) Find f ‘ ( 3 )
4) Find the following three numbers , if they exists:
f(0) and f ‘ (0) and f “ (0)
5) What values of x=c (find out what c equals)
will make the following TRUE?
.......A) f ( c ) = 0 …
(c is the ROOT or ZERO or X-INTERCEPT of the curve.)
.......B) f ‘ (c) = 0 …
(HORIZONTAL TANGENT lines will occur at x = c.)
.......D) f “ (c) = 0 …
(It is very possible that
a POINT of INFLECTION will occur at x = c.)
6) Find the Equation of the NORMAL LINE
that passes thru the point where x = 2 on the curve f(x).
7) For what values of x will the curve
have VERTICAL TANGENT LINES?
8) On what INTERVALS of x will the curve
be INCREASING?
9) On what INTERVALS of x will the curve
be CONCAVED UP?
10) What are the INFLECTION POINTS for the function?
(For what values of x will the curve
CHANGE its CONCAVITY?)

derivative styles uv v/v u to the nth

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Evaluating Numerical Expressions - PEMDAS

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P E MD AS (Please Excuse My Dear Aunt Sally)
"ALWAYS MOVING LEFT TO RIGHT"
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1st) Parenthesis
2nd) Exponents
3rd) * and /
(It maybe helpful to change Divide to Mult. "the INVERSE")
4th) + and -  (change Subt.  to Add "the OPPOSITE")
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48/8*3 is 6*3 which is 18    ...  Do NOT do the Mult. first 48 / 24

(The process does not say to MULT. and THEN DIVIDE.
     It gives them equal billing as you move Left-to-Right!
       Division is really Multiplication of the RECIPROCAL,
        "MD" could just be considered MULTIPLICATION only.)
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Subtraction becomes Additon of the OPPOSITE:
-8 - (-2)  NEGATIVE 8 Minus  Negative 2
Becomes:
Negative 8 PLUS Positive 2     ...      -8 + 2    =    -6
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Simplify Arithmetic Expressions

stretching and shrinking graphs

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Combining Functions

Polar Coordinates

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Where do two lines cross?

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LINE INTERSECTION PROBLEMS:

Compound Interest and "e"

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Compound Interest and "e=2.7182818..."

Derivatives - What f, and f ' and f " can do for you!

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