APPLICATION OF DIFFERENTIATION ; RATE OF CHANGE, EQUATION OF MOTION , MAXIMUM AND MINIMUM POINTS AND VALUES

Rate of Change

If y = f(x),  can sometimes be interpreted as the rate at which y is changing with respect to x. if y increases as x increases,  0, while if y decreases as x increases,  0.

Example

The radius of a circle is increases a t the rate of 0.01cm/s. find the rate at which the area is increasing when the radius of the circle is 5cm.

Example

Water is leaking from a hemisphere bowl of radius 20cm at the rate of 0.5cm3/s. Find the rate at which the surface area of the water is decreasing when the water level is half-way from the top.

MAXIMUM AND MINIMUM POINTS  The points on a curve at which  = 0, are called Stationary Points.

Stationary points fall into three major categories:

(a) Those in which  changes sign from positives through zero to negative. These are called maximum points.

(b)   Those in which  changes sign from negative through zero to positive. These are called minimum points.

(c)   Those in which the sign of  is not changed in the immediate neighborhood of the stationary points. These are called points of inflexion.

The terms maximum and minimum points are used in the local sense and not in the absolute sense.

Fig. 10.12 shows a part of the curve y = f (x). There is a minimum at x = b

At x = b, f1 (b) =0

At x = b, f1 (b) <0

At x = b+, f1 (b+) <0

So a pint on a curve is a minimum at x = a.

If: (i) f (a) = 0      (ii) f’ (a) <0

(iii)    f(a+)>0

shows part of the curve y= f(x). We observe that

f(c) = 0

f(c) < 0

f(c+) < 0

The point x = c is a point of inflexion. Similarly, fig. 10.13(b) shows parts of the curve y = f (x)

We observe that

f(d) = 0

f(d) >0

f ’(d+) >0

The point x = d is a point of inflexion.

A maximum point, a minimum point and a point of inflexion are all stationary points. Both maximum and minimum points are called turning points. A point of inflexion however, is not a turning point.

Example

Find the stationary points in each of the following curves whose equations are:

(a)      y  = x3+ x-3x + 4

3

(b)      y  = x4 +4 x3 – 2x2 – 16x + 1

4     3

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