In the previous lesson, we have learnt that the slope of a line is the tangent of the angle which the is theqline makes with the positive direction of x-axis. It is denoted by the letter ‘m’. Thus, if .qangle which a line makes with the positive direction of x-axis, then m is given by tan.
After seen this video, you will be able to :
● find rate of change of quantities
● find approximate value of functions
● define tangent and normal to a curve (graph of a function) at a point;
● find equations of tangents and normals to a curve under given conditions;
● define monotonic (increasing and decreasing) functions;
● define the points of maximum and minimum values as well as local maxima and local minima of a function from the graph;
● establish the working rule for finding the maxima and minima of a function using the first and the second derivatives of the function; and
● work out simple problems on maxima and minima.
EXPECTED BACKGROUND KNOWLEDGE
● Knowledge of coordinate geometry and
● Concept of tangent and normal to a curve
● Concept of diferential coefficient of various functions
● Geometrical meaning of derivative of a function at a point
● Solution of equetions and the inequations.
RATE OF CHANGE OF QUANTITIES
Solution : Let the foot of the ladder be at a distance x metres from the wall and y metres be the height of the ladder at any time t, then
CHECK YOUR PROGRESS 29.1
1. The side of a square sheet is increasing at rate of 4 cm per minute. At what rate is the area increasing when the side is 8 cm long?
2. An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 10 cm long.
3. Find the rate of change of the area of a circle with respect to its radius when the radius is 6 cm.
4. The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its surface area, when the radius is 7 cm.
5. Find the rate of change of the volume of a cube with respect to its edge when the edge is 5 cm.
In this section, we shall give a meaning to the symbols dx and dy in such a way that the original meaning of the symbol dy/dx coincides with the quotient when dy is divided by dx.
y be theDx be a small change in x and let DLet y = f(x) be a function of x and corresponding change in y. Then,
|The formula is very useful in the calculation of small change (or errors) in dependent variable corresponding to small change ( or errors) in the independent variable.|
SOME IMPORTANT TERMS
CHECK YOUR PROGRESS 29.2
SLOPE OF TANGENT AND NORMAL
CHECK YOUR PROGRESS 29.3
EQUATIONS OF TANGENT AND NORMAL TO A CURVE
CHECK YOUR PROGRESS 29.5
LANGRANGE’S MEAN VALUE THEOREM
This theorem improves the result of Rolle’s Theorem saying that it is not necessary that tangent may be parallel to x-axis. This theorem says that the tangent is parallel to the line joining the end points of the curve. In other words, this theorem says that there always exists a point on the graph, where the tangent is parallel to the line joining the end-points of the graph.
29.6.1 Mathematical Formulation of the Theorem
(a) f is continuous on [a, b], and
(b) f is differentiable in (a, b)
(c) f (b)¹ f (a)
then there exists a point c in the open interval (a, b) such that
CHECK YOUR PROGRESS 29.6
INCREASING AND DECREASING FUNCTIONS
You have already seen the common trends of an increasing or a decreasing function. Here we will try to establish the condition for a function to be an increasing or a decreasing.
(Let a function f x be defined over the closed interval [a,b].)
RELATION BETWEEN THE SIGN OF THE DERIVATIVE AND MONOTONICITY OF FUNCTION
Consider a function whose curve is shown in the Fig. 29.5
CHECK YOUR PROGRESS 29.8
MAXIMUM AND MINIMUM VALUES OF A FUNCTION
We have seen the graph of a continuous function. It increases and decreases alternatively. If the value of a continious function increases upto a certain point then begins to decrease, then this point is called point of maximum and corresponding value at that point is called maximum value of the function. A stage comes when it again changes from decreasing to increasing . If the value of a continuous function decreases to a certain point and then begins to increase, then value at that point is called minimum value of the function and the point is called point of minimum.
Fig. 29.6 shows that a function may have more than one maximum or minimum values. So, for continuous function we have maximum (minimum) value in an interval and these values are not absolute maximum (minimum) of the function. For this reason, we sometimes call them as local maxima or local minima.
A function f (x) is said to have a maximum or a local maximum at the point x = a where a – b < a < for all sufficiently small positive b.) ( ± ³ ) (a + b (See Fig. 29.7), if f a f a b
CONDITIONS FOR MAXIMUM OR MINIMUM
We know that derivative of a function is positive when the function is increasing and the derivative is negative when the function is decreasing. We shall apply this result to find the condition for maximum or a function to have a minimum. Refer to Fig. 25.6, points B,D, F are points of maxima and points A,C,E are points of minima.
, but on the right of B, the> ) (Now, on the left of B, the function is increasing and so f ‘ x 0 . This can be achieved only when f ‘(x) becomes< ) (function is decreasing and, therefore, f ‘ x 0 zero somewhere in betwen. We can rewrite this as follows :
A function f (x) has a maximum value at a point if (i) f ‘(x) = 0 and (ii) f ‘(x) changes sign from positive to negative in the neighbourhood of the point at which f ‘(x)=0 (points taken from left to right).
Now, on the left of C (See Fig. 29.6), function is decreasing and f ‘(x) therefore, is negative and on the right of C, f (x) is increasing and so f ‘(x) is positive. Once again f ‘(x) will be zero before having positive values. We rewrite this as follows:
A function f (x) has a minimum value at a point if (i) f ‘(x)=0, and (ii) f ‘(x) changes sign from negative to positive in the neighbourhood of the point at which f ‘(x) = 0.
We should note here that f ‘(x) = 0 is necessary condition and is not a sufficient condition for maxima or minima to exist. We can have a function which is increasing, then constant and then again increasing function. In this case, f ‘(x) does not change sign.The value for which f ‘(x) = 0 is not a point of maxima or minima. Such point is called point of inflexion.
(For example, for the function 3 is the point of= =f x x ,x 0 ) (inflexion as 2 does not change sign as x passes=f ‘ x 3x through 0. f'(x) is positive on both sides of the value ‘0’ (tangents make acute angles with x-axis) (See Fig. 29.9). )
Hence 3 has a point of inflexion at x = 0.=f x x The points where f ‘(x) = 0 are called stationary points as the rate of change of the function is zero there. Thus points of maxima and minima are stationary points.
METHOD OF FINDING MAXIMA OR MINIMA
We have arrived at the method of finding the maxima or minima of a function. It is as follows :
- Find f'(x)
- Put f ‘(x)=0 and find stationary points
- Consider the sign of f ‘(x) in the neighbourhood of stationary points. If it changes sign from +ve to –ve, then f (x) has maximum value at that point and if f ‘(x) changes sign from –ve to +ve, then f (x) has minimum value at that point.
- If f ‘(x) does not change sign in the neighbourhood of a point then it is a point of inflexion.
CHECK YOUR PROGRESS 29.9
Find all points of local maxima and local minima of the following functions. Also , find the maxima and minima at such points.
USE OF SECOND DERIVATIVE FOR DETERMINATION OF MAXIMUM AND MINIMUM VALUES OFA FUNCTION
Example 29.35: Find the local minimum of the following function :
CHECK YOUR PROGRESS 29.10
Find local maximum and local minimum for each of the following functions using second order derivatives.
APPLICATIONS OF MAXIMA AND MINIMA TO PRACTICAL PROBLEMSAPPLICATIONS OF MAXIMA AND MINIMA TO PRACTICAL PROBLEMS
The application of derivative is a powerful tool for solving problems that call for minimising or maximising a function. In order to solve such problems, we follow the steps in the following order :
- Frame the function in terms of variables discussed in the data.
- With the help of the given conditions, express the function in terms of a single variable.
- Lastly, apply conditions of maxima or minima as discussed earlier.
Example 29.39: Find two positive real numbers whose sum is 70 and their product is maximum.
Example 29.43: Show that the height of a closed right circular cylinder of a given volume and least surface is equal to its diameter.
Solution : Let V be the volume, r the radius and h the height of the cylinder.
Example 29.47: Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Solution : Let h be the height and R the radius of the base of the inscribed cylinder. Let V be the volume of the cylinder.