The French method of drawing curves is very systematic:

*“Pratique de l’etude d’une fonction” *

Let * f* be the function represented by the curve

**C**Steps:

1. Simplify ** f(x)**. Determine the

**Domain of definition**(

**D**) of

**;**

*f*2. Determine the

**sub-domain E**of D, taking into account of the

**periodicity**(eg. cos, sin, etc) and

**symmetry**of

*;*

**f**3. Study the

**Continuity**of

**;**

*f*4. Study the

**derivative**of

*and determine*

**f****f'(x)**;

5. Find the

**limits**of

*within the boundary of the intervals in E;*

**f**6. Construct the Table of Variation;

7. Study the

**infinite**branches;

8. Study the remarkable points: point of

**inflection**,

**intersection**points with the X and Y axes;

9. Draw the representative curve

**C**.

Example:

**Step 1**: Determine the Domain of Definition D

D = R* = R – {0}

**Step 2**: There is no Periodicity and Symmetry of f

E = D = R*

[See Note below for Periodic and Symmetric example]

**Step 3:** Continuity of f

The function f is the quotient of 2 polynomial functions, therefore f is **differentiable**

=> **f** is * continuous* in

[See previous post CID Relation]

**Step 4**: Determine **f’**

Therefore* f’* has the

**same sign**as

**Step 5a**: Limit at x=0

Therefore,

**Step 5b**: Limit at

**Step 5c**: Limit at

Similarly,

**Step 6**: Construct the Table of Variation

**Step 7: **Study the infinite branches

7a)

=> **y-axis** is the **asymptote**

7b)

,

=>

,

=> **y= x** is another **asymptote**

=> The curve** C** is above the **asymptote y=x**

**Step 8**: Study the remarkable points: **intersection** points with x-axis

**Step 9**: Draw the representative curve **C** of **f**.

Note:

D = R

g(x) is periodic of 2π => E = **[0 , 2π]**

=> g(x) is symmetric with respect to the origin point O

We can restrict our study of g(x) in E = **[0,π]**

=> g(x) is symmetric w.r.t. to the equation **x= π/2**

Finally, we can further restrict our study of g(x) in E = **[0, π/2]**

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Reblogged this on Singapore Maths Tuition.

Reblogged this on Singapore Maths Tuition.