Application of the derivative to the analysis of functions . With the concept of derivative some local properties of the functions can be studied, the study of these characteristics will facilitate the graphic representation of them.
Summary
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- 1 Growth and decrease of functions in an interval
- 2 Local ends
- 3 Concavity and convexity of a function
- 4 inflection points
- 5 Other applications of the derivative
- 1 Rough calculation of the values of a function
- 6 Source
- 7 See also
Growth and decrease of functions in an interval
- If f is a differentiable function on the interval (a; b) and for each x with a <x <b, f ‘ (x)> 0 holds , then the function f is strictly increasing on the given interval.
- If f is a differentiable function on the interval (a; b) and for each x with a <x <b, f ‘(x) <0 is true, then the function f is strictly decreasingon the given interval.
Example:
Determine the growth and decrease intervals of the function: y = ¹ / 3 x 3 + x 2 + 1
Resolution
As y ‘= x 2 + 2x = x (x + 2)
The sign of the expression x (x + 2) is analyzed
y ‘is positive if x <-2 or if x> 0
y ‘is negative if -2 <x <0
Therefore the function is strictly increasing in the intervals (- ; -2) and (0; – ) and decreasing in the interval (-2; 0)
Local ends
A point x 0 is a local extreme (maximum or minimum) of a function f, if the value f (x 0 ) is greater (maximum) or less (minimum) than all the values that the function takes in an interval of the type ( x 0 -µ; x 0 + µ).
Theorem
For a differentiable function at x 0 to have a local end at x 0, it is necessary that f ‘(x 0 ) = 0 be fulfilled .
Since growth is determined by the sign of the derivative we have:
x 0 is a point of
- Local maximum if f ‘(x) goes from positive to negative.
- Local minimum if f ‘(x) goes from negative to positive.
Example
Find the local extremes of the function: y = x 3 -12x-4
Resolution
Since y ‘= 3x 2 -12 = 3 (x 2 -4) = 0
The zeros of y ‘are x = 2 and x = -2. When analyzing the sign of y ‘we find
y ‘> 0 in (- ; -2) and (2; + )
y ‘<0 at (-2; 2), then at x = -2 y’ goes from positive to negative values, it is a maximum that is y max = f (-2) = 12.
At x = 2 y ‘goes from negative to positive values, it is a maximum that is y min = f (2) = – 20.
Second derivative theorem
Let f be a function twice differentiable at x 0 .
If f ‘(x 0 ) = 0 and f’ ‘(x 0 ) 0 then f has a local end at x 0
If f “(x 0 )> 0 the extreme is a local minimum.
If f “(x 0 ) <0 the extreme is a local maximum.
Example:
Use the criterion of the second derivative to find the extremes of the indicated function:
f (x) = x 3 – 6x 2 – 15x
Solution:
f´ (x) = 3 x 2 – 12 x – 15 = 0 ⇒ Critical points: x 1 = -1 and x 2 = 5
f´´ (x) = 6x – 12 ⇒ f ´´ (-1) = -18 <0 ⇒ in x 1 = -1 there is a maximum of f.
⇒ f´´ (5) = 18> 0 ⇒ at x 2 = 5 we have a minimum of f.
Concavity and convexity of a function
A function is concave if a unit vector is fixed on the positive semi-axis OY, this vector is on the same half plane (determined by the tangent lines to the function) as the function. Otherwise (different half planes) it is called convex.
Concavity and convexity analytical conditions
If f “(x)> 0 on an interval (a, b), then the function f (x) is concave on the interval (a, b).
If f “(x) <0 on an interval (a, b), then the function f (x) is convex on the interval (a, b).
Example: f (x) = x 3 -3x 2 + 6x-6
f ‘(x) = 3x 2 -6x + 6
f “(x) = 6x-6
6x-6> 0 ⇒ x> 1 concave (1; )
6x-6 <0 ⇒ x <1 convex (- ; 1)
Turning points
A turning point is a point where the values of x of a continuous function pass from one type of concavity to another.
Theorem
Let y = f (x) be the equation of a function.
If f “(a) = 0 of“ (a) does not exist, and the derivative f “(x) changes sign when passing through the value of x = a, then the point of the abscissa function x = a is a turning point.
Example: f (x) = x 3 -3x 2 + 6x-6
f ‘(x) = 3x 2 -6x + 6
f “(x) = 6x-6
6x-6 = 0 ⇒ x = 1
The point x = 1 is an inflection point, since before x = 1 the second derivative is negative (convex) and after x = 1 it is positive (concave).
Other applications of the derivative
Rough calculation of the values of a function
Introducing the derivation allows us to make more accurate approximate calculations for the derivable functions, provided x is chosen small enough and the following basic formula is used.
f (x 0 + x) ≈ f (x 0 ) + xf ‘(x 0 )
Example
Calculate approximately
Resolution
To apply the formula, we must find an x 0 close to 38 in which it can be calculated accurately ; in this case x 0 = 36 is convenient.
