1. 9. Point of Inflexion
Definition: A point of inflexion or a point of inflection or inflexional point or inflectional point (x0,y0) on a curve y=f(x) is a continuous point at which the function f(x) changes from convex to concave or vice versa.

2. How many inflexional points are there?+ - 1. x
x< 0
0<x<1
x>1 y” - + 2

3. 10. Asymptotes to a Curve

4. How many kinds of asymptotes are there?
A) Horizontal asymptote y=a
B) Vertical asymptote x=b
C) Oblique asymptote y= ax + b

5. Definition of an asymptote to a curve
A straight line is an asymptote to a curve if and only if the perpendicular distance from a variable point on the curve to the line approaches to zero as a limit when the point tends to infinity along the curve on both sides or one side of the curve.

6. Theorem 10.1 y=ax+b is an oblique asymptote to y=f(x) iff
Proof

7. Thm 10.2
Proof

8. Examples of finding asymptotes

9. The oblique asymptote is y= the quotient when P(x) is divided by Q(x)
Oblique asymptote for special rational function f(x)=P(x)/Q(x), where deg(P(x)) = deg(Q(x))+1
The oblique asymptote is
y= the quotient when P(x) is
divided by Q(x)

10. Procedures of sketching curves e.g.
Domain of f(x)
Symmetry of f(x):periodicity, even or odd
Find f’(x) and f ”(x)
Monotonicity and Extrema of f(x)
Convexity and Inflexional points of f(x)
Asymptotes
Special points e.g. intercepts

11. Sketch of