Ch 1. Limits     Multimedia Engineering Math
Limit of a
Sequence		 Limit of a
Function		 Limit
Laws		 Continuity Rate
Change		      
Math
 Rate Change Case Intro Theory Case Solution  
 Chapter
 1. Limits
 2. Derivatives I
 3. Derivatives II
 4. Mean Value
 5. Curve Sketching
 6. Integrals
 7. Inverse Functions
 8. Integration Tech.
 9. Integrate App.
 10. Parametric Eqs.
 11. Polar Coord.
 12. Series
 
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Author(s):
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Kurt Gramoll
 
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MATHEMATICS - THEORY
    Slope of a line

Slope of a Line
  

The slope of a line is denoted by m and can be determined with two points: A(x1,y1) and B(x2, y2).

     

It can also be determined by:

     

in which α is the smallest positive angle from the x axis to the line. When the line is horizontal, m = 0 and when the line is vertical, m is infinity or not defined.
   

 
    Tangent

Slope of a Curve
  

Suppose curve C (f(x)) is a function of x, the tangent line of the curve C at point A is desired.

Now, consider B point where x2 is not equal to x1, the slope of AB is:

     
     

Tangent of a Curve
  

Let point B approaches A along the curve C by letting x2 approaches x1, when x2 - x1 is small enough, the slope of AB will overlap with its tangent line at point A.Thus, the tangent at point A can be written as:

     
     
    Tangent Example - Velocity
   

Velocity is a good example of a tangent line. Normally, the distance s is a function of time t. The velocity of a time interval is

     
     

Velocity of any Function
  

On a graph of distance plotted as a function of time, the tangent (instantaneous velocity) is:

     

Therefore, the instantaneous velocity at time t, is equal to the slope of the tangent line at point A.