math
10th Grade - Study of a function

10th Grade lesson

4 - Study of functions


To study a function

1. Calculate the derivative of the function.
2. Determine the sign of the derivative.
3. Calculate the limits of the function at the bounds of its domain and the monotony of the function for the values of x at which the sign of f' changes. Then you can draw its monotony table.

Examples

*** Study the monotony of derivative.

1. Calculate the derivative. Here derivative.

2. Determine the sign of the derivative : derivative function study, then f' is positive whenever signe derivee.

3. Calculate the limits of f at the bounds of its domain. Here, limit calculation.
function limit

There is an indeterminate form for the calculation of the limit at infinite. Factorize by the term of highest degree :
limit

function limit

Then calculate f(1) : math.
Then draw the monotony table this way :

math


*** Study the monotony of math

To calculate the derivative, suppose derivative function calculation and derivative function calculation. Then derivative function calculation and derivative function calculation. Hence :
derivative function calculation

Here the study of the sign of the derivative is rather quick because the numerator is always positive : delta calculation and 5 > 0 then the parabola is always above the X-axis, and the denominator too (a square is always positive, here notice the interest not to expand the denominator - previous chapter - ). f is not defined at x = -1 and at x = 1 then interval. You can do limit calculations for the limits in minus infinity and plus infinity, factorize up and down by x squared and simplify, and for the limits at local derivative, local derivative, local derivative, and local derivative the result is always infinity, + or -, depending on the sign of sign.
Here is the table :

variation table

Equation of the tangent

Often, in problems, you'll be asked to give the equation of the tangent to a graph of function f at some point x = a, that means to give the equation of the red line, that intersects the graph of f at point M of abscissa x = a.

function graph


The red line is a line, its equation is of the form line equation. According to lesson about the derivatives, the slope of the tangent at a point is the derivative of f at this point. Then the equation of the red line is : line equation. Because the point point coordinates belongs to the line, its coordinates verify the equation of the line, then line equation. By replacing the value of p in the equation line equation, we finally obtain the general formula : tangent equation
To calculate the equation of the tangent of a function f at x = 2, you just have to calculate f'(2), f(2), and replace the results in the formula above.



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Study of functions

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