The instantaneous rate of change or the derivative
         Continuity and discontinuity of a function
         Vertical, horizontal and oblique or slant asymptotes
The instantaneous rate of change or the derivative
Thus, if the slope or gradient m of the secant line passing through the points (x1, ƒ(x1)) and (x2, ƒ(x2)) of the graph of a function is positive, the function is increasing (going up), as shows the figure below. 
where,   x2 - x1> 0
 
 
Since the difference x2 - x1 is always positive, when the function is decreasing (going down), the slope will be negative.
The ratio of the rise and the run, called the difference quotient, that equals the value of the tangent of the angle between the direction of the secant line and x-axis, becomes the slope (gradient) of the tangent line as the difference Dx tends to zero, and is called the instantaneous rate of change or the derivative at the point of the function.
For a given function ƒ and point (x1, ƒ (x1)), the derivative of ƒ at x = x1 is the slope of the tangent line through the point (x1, ƒ (x1)), i.e.,  f '(x1) = tan at .
The gradient of a curve at a point on its graph, expressed as the slope of the tangent line at that point, represents the rate of change of the value of the function and is called derivative of the function at the point, written
y' = dy/dx =  f '(x)
Continuity and discontinuity
A function that has no sudden changes in value as the variable increases or decreases smoothly is called continuous function.
Or more formally, a real function  y = ƒ (x) is continuous at a point a if the limit of ƒ (x) as x approaches a is ƒ (a).
If a function does not satisfy this condition at a point it is said to be discontinuous, or to have a discontinuity at that point.
Vertical, horizontal and oblique or slant asymptotes
A line whose distance from a curve decreases to zero as the distance from the origin increases without the limit is called the asymptote.
The definition actually requires that an asymptote be the tangent to the curve at infinity. Thus, the asymptote is a line that the curve approaches but does not cross.
Vertical asymptote
The line x = a is a vertical asymptote of a function ƒ if ƒ (x) approaches infinity (or negative infinity) as x approaches a from the left or right.
Horizontal asymptote
The line y = c is a horizontal asymptote of a function ƒ if ƒ (x) approaches c as x approaches infinity (or negative infinity).
Oblique or slant asymptote
The line  y = mx + c  is a slant or oblique asymptote of a function ƒ if ƒ (x) approaches the line as x approaches infinity (or negative infinity).
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The functions that most likely have asymptotes are rational functions.
So, vertical asymptotes occur when the denominator of the simplified rational function is equal to 0. Note that the simplified rational function has cancelled all factors common to both the numerator and denominator.
The existence of the horizontal asymptote is related to the degrees of both polynomials in the numerator and the denominator of the given rational function.
Horizontal asymptotes occur when either, the degree of the numerator is less then or equal to the degree of the denominator.
In the case when the degree (n) of the numerator is less then the degree (m) of the denominator, the x-axis
y = 0 is the asymptote.
If the degrees of both polynomials, in the numerator and the denominator, are equal then,  y = an/bm  is the horizontal asymptote, written as the ratio of their highest degree term coefficients respectively.
When the degree of the numerator of a rational function is greater than the degree of the denominator, the function has no horizontal asymptote.
A rational function will have a slant (oblique) asymptote if the degree (n) of the numerator is exactly one more than the degree (m) of the denominator that is if  n = m + 1.
Dividing the two polynomials that form a rational function, of which the degree of the numerator pn(x) is exactly one more than the degree of the denominator qm (x), then
pn (x) = Q (x) · qm(x) + R      =>    pn (x)/qm (x) = Q (x) + R/qm (x)
 where, Q (x) = ax + b is the quotient and R/qm (x) is the remainder with constant R.
The quotient Q (x) = ax + b represents the equation of the slant asymptote.
As x approaches infinity (or negative infinity), the remainder R/qm (x) vanishes (tends to zero).
Thus, to find the equation of the slant asymptote, perform the long division and discard the remainder.
The graph of a rational function will never cross its vertical asymptote, but may cross its horizontal or slant asymptote. 
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Example:  Given the rational function sketch its graph.
Solution:  The vertical asymptote can be found by finding the root of the denominator,
                      x + 1 = 0       =>       x =  -1  is the vertical asymptote.
The horizontal asymptote is the ratio of their highest degree term coefficients since the degree of polynomials in the numerator and denominator are equal,
  is the horizontal asymptote.
The graph of the given rational function is translated equilateral (or rectangular) hyperbola shown below.
The rational function of the form
can be rewritten into
so  
where, x0 and y0 are asymptotes and k is constant.
 
Therefore, values of the vertical and the horizontal asymptote correspond to the coordinates of the horizontal and the vertical translation of the source equilateral hyperbola  y = k/x, respectively.
Example:  Given the rational function   sketch its graph.
Solution:  The vertical asymptote can be found by finding the root of the denominator, 
                      x + 2 = 0       =>      x =  -2  is the vertical asymptote.
Since the degree of the numerator is exactly one more than the degree of the denominator the given rational function has the slant asymptote.
By dividing the numerator by the denominator
obtained is the slant asymptote  y = x 
and the remainder  3/(x + 2) that vanishes as x approaches positive or negative infinity.
 
Calculus contents A
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