Calculus I
Limits
Derivative
Exercices
Applications
Marginal analysis
© The scientific sentence. 2010

Calculus I: Graphing a function
1. How to graph a function
Let f a function. To study f, we proceed by the following steps:
 Find the domain of the function,
 Set the continuity of this function,
 Determine the parity of the function,
 Determine its asymptotes,
 Derive is f(x) and determine the critical numbers
and the relative extrema of the function.
 Set f"(x) and the numbers of transition and the
inflection points.
 Construct the signvariation table of the function
including the variation of f and its concavity.
 Draw the graph.
2. Example:
f(x) = (x^{2} + 3)/(x^{2}  9)
 Domain D = R / { 3, + 3}
 f is discontinuous at x =  3 and x = + 3,
lim f(x) = 12/0^{} =  ∞
x →  3^{}
lim f(x) = 12/0^{+} = + ∞
x →  3^{+}
lim f(x) = 12/0^{} =  ∞
x → + 3^{}
lim f(x) = 12/0^{+} = + ∞
x → + 3^{+}
 f is even because f( x) = f(x)
The interval of symmetry is [0, + ∞[

The asymptotes are:
Vertical: x =  3 and x = + 3
Horizontal:
lim f(x) = lim (x^{2}/x^{2}) = 1
x → ± ∞
 f'(x) =  24 x /(x^{2}  9)^{2}
x = 0 f(0) =  1/3
x = 0 exists in the domain D. So this point
is a critical point. The point (0,  1/3) is an extremum.
 f"(x) = (72 x^{2} + 216)/(x^{2}  9)^{3}
f"(x) = 0 has no solutions. So there are no numbers of transition
or inflection points.
f "(0) = 216/( 9)^{3} < 0 . So
the point x = 0 is a relative maximum of f and the
concavity is downward (convex).


3. Exercises
A homographic function f is defined as:
f(x) = (ax + b)/(cx + d)
where a, b, c, and d are real constants.
 D = R \ { d/c}
 f is discontinuous at x =  d/c
lim f(x) = 12/0^{±} = ± ∞
x → ( d/c)^{±}
The line x =  d/c is a vertical asymptote.
lim f(x) = a/c
x → ± ∞
The line y = a/c is a horizontal asymptote.
 The function is neither even nor odd, so
we study the function in the whole domain D.

There is no oblique asymptote because the function has
already a horizontal asymptote.

f '(x) = (ad  bc)/(cx + d)^{2}
has no solutions.
So no critic points, then
no relative extrema.
The function is increasing if (ad  bc) >0,
and decreasing if (ad  bc) <0.
 f"(x) =  2 c (ad  bc)(cx + d)/(cx + d)^{4}
f"(x) = o when x =  d/c
This point x =  d/cD, so it is not a
transition number, then not an inflection point.
Using the division of polynomials, we can write:
f(x) = (ax + b)/(cx + d) = (a/c) + (1/c)(bc  ad)/(cx + d)
and we see y = a/c is a particular oblique asymptote,
that is a horizontal asymptote.
We can also remark that this function f(x) is transformed
by a translation from the function f_{o}(x) = 1/x,
with (a/c) vertically, and ( d/c) horizontally. Hence the word
homograph i.e. same graph.
 Construct the signvariation table of the function
including the variation of f and its concavity.
 Draw the graph.

