To study a function1. 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 .
1. Calculate the derivative. Here .
2. Determine the sign of the derivative : , then f' is positive whenever .
3. Calculate the limits of f at the bounds of its domain. Here, .
There is an indeterminate form for the calculation of the limit at . Factorize by the term of highest degree :
Then calculate f(1) : .
Then draw the monotony table this way :
*** Study the monotony of
To calculate the derivative, suppose and . Then and . Hence :
Here the study of the sign of the derivative is rather quick because the numerator is always positive : 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 . 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 , , , and the result is always infinity, + or -, depending on the sign of .
Here is the table :
Equation of the tangentOften, 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.
The red line is a line, its equation is of the form . 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 : . Because the point belongs to the line, its coordinates verify the equation of the line, then . By replacing the value of p in the equation , we finally obtain the general formula :
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.