Application of Derivatives

Application of Derivative
Derivative is primarily used to determine rates of changes of quantities. Derivative leads to the determination of slope of the tangent to a curve at a given point. Derivative also helps in locating the turning points of the graph of a function.


Geometrical Meaning of the Derivative:
Let y = f (x) represent a given curve, and P (x, y) and Q(x + x, y + y) be two distinct and neighbouring points on the curve. Let the chord PQ, joining P and Q, make an angle with the x-axis and the tangent to the curve y = f(x) at the point (x, y) make an angle with the positive x -axis. The slope of the chord PQ is given by
tan =

.



Tangent and Normal to a Curve:

PT is the tangent to the curve y = f(x) at the point P(x₁, y₁). PN is the normal to the curve at P. The normal to the curve at point P is perpendicular to the tangent to the curve at P.

The slope of the tangent at P(x₁, y₁) is,

.


The slope of the normal at P(x₁, y₁) is,



.

Hence the equation of the tangent PT is, , and the equation of the normal PN is,

= (x - x₁) + (y - y₁) = 0.

The equations of the tangent and the normal are

and .



Angle Between two Curves

The angle between two curves (or the angle of intersection of two curves) is defined as the angle between the two tangents at their point of intersection. As the figure shows, , the angle

between the two curves, is given by = ₁ -₂ tan = tan (₁ -₂)

=

where tan₁ = '(x₁) and tany₂ = g'(x₁).



Two curves are said to cut each other orthogonally if the angle between them is a right angle, that is, if = 90⁰ , in which case we will have, tan₁ tan₂ = -1.

Two curves touch each other if the angle between the tangents to the curves at the point of intersection is 0^o , in which case we will have, tan₁ = tan₂ .



Monotonocity

Let y = f (x) be a given function with 'D' as it's domain. Let .D₁ D



Increasing Function:

1.f (x) is said to be increasing in D₁ if for every

x₁, x₂ D₁, x₁ > x₂ f(x₁) > f (x₂). It means that the value of f (x) will keep on increasing with an increase in the value of x. Refer to fig.2.





Non-Decreasing Function:

1. f (x) is said to be non-decreasing in D₁ if for every x₁ , x₂ D₁ , x₁ > x₂

f (x₁) f (x₂). It means that the value of

f (x) would never decrease with an increase in the value of x. Refer to fig.3.




Decreasing Function:

1. f (x) is said to be decreasing in D₁ if for every

x₁, x₂ D₁, x₁ > x₂

f (x₁) < f (x₂). It means that the

value of f (x) would decrease with an increase in the value of x. Refer to fig.4.





Non-Increasing Function:

1. f (x) is said to be non-increasing in D₁ if for every x₁, x₂ D₁, x₁ > x₂

f (x₁) f (x₂). It means that the value of f (x) would never increase with an increase in the value of x. Refer to fig. 5.





Note:
A function which is either increasing or decreasing in the entire interval D₁ is called a monotonic function.



Basic Theorems:

Let, f be a function that is continuous in [a, b] and differentiable in (a, b). Then

(i) f(x) is a non-decreasing function in [a, b] if f '(x) 0 in (a, b) ;

(ii) f(x) is an increasing function in [a, b] if f '(x) > 0 in (a, b) ;

(iii) f(x) is a non-increasing function in [a, b] if f '(x) 0 in (a, b) ;

(iv) f(x) is a decreasing function in [a, b] if f '(x) < 0 in (a, b).



Remarks:

(i) Let x₁, x₂, (x₁ < x₂) be any two points in [a, b]. Since the function is differentiable and continuous in [a, b], f(x) satisfies the conditions of Lagrange's Mean value theorem on [a, b].

Hence on [x₁, x₂],f(x₂) - f(x₁) = (x₂ - x₁) f'(c), x₁ < c < x₂. Since, x₂ - x₁ > 0, f(x₂) > f(x₁) if f'(c) > 0 and f(x₂) < f (x₁) if f'(c) < 0.

This result is true for any two points x₁ and x₂ in [a, b] and c (x₁, x₂). Since c is arbitrary, we conclude that if f' (x) > 0 in [a, b], then f (x) is an increasing function and if f'(x) < 0 in

[a, b], then f(x) is a decreasing function.

(ii) Ifand points which make equal to zero (in between (a, b) don't form an interval, then f (x) would be increasing in [a, b].

(iii) Ifand points which make equal to zero (in between (a, b) don't form an interval, f (x) would be decreasing in [a, b].

(iv) If f (0) = 0 and then f (x) 0 x (- , 0) and f (x) 0 x (0, ).

(v) If f (0) = 0 and then f(x) 0 x (-, 0) and f (x) 0 x (0, ).

(vi) The points for which is equal to zero or doesn't exist are called critical points. Here it should also be noted that critical points are the interior points of an interval.

(vii) The stationary points are the points where = 0 in the domain.

If f' (x) is continuous, then f (x) can go from positive values to negative values or from negative values to positive only by going through the value zero. The value of x for which f' (x) = 0

are also called the turning points. At a stationary or a turning point, the tangent to the curve y = f (x) is parallel to the x-axis. On the left and right of a turning point, tangents to the curve

have different direction.

(viii) A function f (x) is said be to increasing or decreasing at a point x₀, if there exists an interval (x₀ - h, x₀ + h), h > 0 around x₀, in which the function is increasing or decreasing

respectively.




Maxima and Minima

Let y = f (x) be a function which is defined on an open interval (a, b) and let c be a point in this interval. Let I (c - h, c + h), h > 0, be an infinitesimal interval about c.

f(x) is said to have a local or relative maximum at x = c, if there exists a neighbourhood

(c - h, c + h), (contained in the domain of f) , of c such that f(c) f(x) for every x (c - h, c + h) or f(c) > f(x) x (c - h, c)(c, c + h).

f(x) is said to have a local or relative minimum at x = c if there exists a neighborhood (c - h, c+ h), (contained in the domain of f), of c such that f(c) f(x) for every x (c - h, c +

h).f(x) is said to have relative or local extremum at x = c if it has relative maximum or relative minimum at x = c.

If f(x) has a local maximum (minimum) at c, then f(c) is called a local maximum (minimum) value of f. The points of local maximum and minimum are called critical points or stationary

points. The values of the function corresponding to these points are called the extreme values of the function.






Theorem:

If f(x) has a local extreme value at x = c then either f ' (c) = 0 or f '(c) does not exist.


Note:
A function may have many local maxima and/or local minima.

The local maximum (or the local minimum) value of a function is not the largest (or the smallest) value of the function. At a point of local maximum (or local minimum) a function has the

largest (or the smallest) value in comparison with the values in the neighbourhood of that point.



Tests for Maxima and Minima:

If f '(c) = 0, then we have 3 tests to decide whether function f(x) has local maxima or local minima or neither at x = c.


First Derivative Test:

Let f(x) be continuous in some neighbourhood (c - h, c + h) of c. Then

(i) f(x) has a local maximum at x = c if ;

(a) x = c is a critical point of f(x);

(b) f '(x) > 0 in (c - h, c), and (c) f ' (x) < 0 in (c, c + h).

Here, in moving from left to right through the critical point c, f '(x) changes sign from plus to minus.

(ii) f(x) has a local minimum at x = c if ;

(a) x = c is a critical point of f(x); (b) f '(x) < 0 in (c - h, c), and

(c) f ' (x) > 0 in (c, c + h).

Here, in moving from left to right through the critical point c, f '(x) changes sign from minus to plus.

(iii) If f '(x) does not change sign in moving through c, then there is neither a maximum nor a minimum at x = c.



Remarks:

If f '(c) does not exist then the following method is adopted in deciding whether f(x) has a local maximum/minimum at x = c or not.


Case 1:

Let f'_-(c) and f'₊(c) exist, be non zero and unequal with A = f'_-- (c), B = f'₊(c);

(a) If A > 0 and B < 0, then f(x) has local maximum at x = c.

(b) If A < 0 and B > 0, then f(x) has local minimum at x = c.

(c) If A and B both are of the same sign then f(x) does not have any local maxima or local minima at x = c.


Case 2:

Let one of the derivatives (either left or right) be zero at x = c and the other

non-zero. If f'_-(c) = 0 and f'₊(c) 0, we write A = f ' (c - h) and B = f'₊ (c) (h > 0) and follow the procedure discussed in case (1). Similarly for f'_-(c) 0 and f'₊(c) = 0.



Case 3:In case the left or the right or both the derivatives do not exist at x = c, but f(x) is continuous at x = c, we consider the value of the function for x = c - h and x = c + h. If f(c - h) < f(c)

as well as f(c + h) < f(c), then f(x) has a local maximum at x = c. If f(c - h) > f(c) and f(c + h) > f(c), then f(x) has a local minimum at x=c.



Concept of Global Maximum/Minimum:

Let y = f (x) be a given function with domain D. Let [a, b] D. Global maximum/minimum of f (x) in [a, b] is basically the greatest/least value of f (x) in [a, b].

Global maximum and minimum in [a, b] would always occur at critical points of f(x) within [a, b] or at the end points of the interval.



Global Maximum / Minimum in the closed interval [a, b]:

In order to find the global maximum and minimum of f(x) in [a, b], find out all the critical points of f(x) in (a, b). Let c₁, c₂, .....c_n be the different critical points. Find the value of the

function at these critical points. Let f(c₁), f(c₂), .., f(c_n) be the values of the function at the critical points.

Say, M₁ = max {f(a), f(c₁), f(c₂), .., f(c_n), f(b)}

and M₂ = min {f(a), f(c₁), f(c₂), .., f(c_n), f(b)}.

Then M₁ is the greatest value of f (x) in [a, b] and M₂ is the least value of f (x) in [a, b].



Global Maximum / Minimum in the open interval (a, b):

Let y = f (x) be the given function and c₁, c₂, .., .c_n be different critical points of the function in (a, b).

Let M₁ = max {f(c₁), f(c₂), f(c₃) .., f(c_n)}

and M₂ = min {f(c₁), f(c₂), f(c₃) .., f(c_n)}.

Now if f(x) > M₁ (or < M₂), f(x) does not have a global maximum (or a global minimum) in (a, b).

This means that if the limiting values at the end points are greater than M₁ or less than M₂, then f(x) would not have global maximum/minimum in (a, b). On the other hand if

M₁ >f (x) and M₂ <f(x), then M₁ and M₂ are respectively the global maximum and global minimum of f (x) in (a, b).



Second Derivative Test:

Let f be a function such that,

(i) f(x) is continuous in (c - h, c + h) ;

(ii) f '(c) = 0;

(iii) f '' (c) exists.

Then,

(1) f(x) has a local maximum at x = c, if f ''(c) < 0.

(2) f(x) has a local minimum at x = c, if f ''(c) > 0.

The second derivative test does not help when f '(c) = 0 and f ''(c) = 0. In such a situation we shall have to depend upon the following test.



n^th Derivative Test:

Let f be a function such that

(i) f '(c) = f ''(c) = ... = f ^{(n - 1)} (c) = 0; (ii) f⁽ⁿ⁾ (c) 0. Then,

(1) f(x) has a local maximum at x = c ; if n is even and f ⁽ⁿ⁾ (c) < 0.

(2) f(x) has a local minimum at x = c ; if n is even and f ⁽ⁿ⁾ (c) > 0
.
(3) f(x) has no local extreme at x = c ; if n is odd.



Differentials and approximations:

Let y = f (x) be a differentiable function. By definition

< for |x| <.

This can also be written as

.. (1)

where 0 as x 0. When dx is small, . dx is still smaller, and (1) can be approximated by

dy =.


Therefore, when x is increased by a small quantity x, y will increase approximately by f' (x) dx. The expression f' (x) dx is the differential of y and its value depends upon the values

assigned to both x and dx.

We, thus, have f (x + x) - f (x) = dy = f' (x) x f (x + x) = f (x) + f' (x) x.



Curve Sketching

We can analyze the shape of the graph of a given curve of the form f (x, y) = 0, by using one or more of the following information. Such an analysis enables to sketch or trace the graph

of the given curve.

1. Domain and range (or the extent of the curve)

2. Points of intersection with the coordinate axes

3. Symmetry about the coordinates axes i.e. f (x, y) = f (x, - y) or f (x, y) = f (- x, y) or both.

4. Symmetry about the origin i.e. f (x, y) = f (- x, - y).

5. Symmetry about the lines y = x, i.e. f (x, y) = f (y, x) or f (x, y) = f (- y, - x).

6. Intervals of increase and decrease of the curve

7. Points of local maximum and local minimum (turning points)

8. Periodicity. If p is the period of y = f (x), then it is enough to trace the function in the interval [a, a + p]. Then the graph repeats itself on both sides of the interval.