Application of Derivatives - NCERT

1 MATHEMATICS194vWith the Calculus as a key, Mathematics can be successfully appliedto the explanation of the course of Nature. WHITEHEAD Chapter 5, we have learnt how to find derivative of composite functions, inversetrigonometric functions, implicit functions, exponential functions and logarithmic this chapter, we will study applications of the derivative in various disciplines, , inengineering, science, social science, and many other fields. For instance, we will learnhow the derivative can be used (i) to determine rate of change of quantities, (ii) to findthe equations of tangent and normal to a curve at a point, (iii) to find turning points onthe graph of a function which in turn will help us to locate points at which largest orsmallest value (locally) of a function occurs. We will also use derivative to find intervalson which a function is increasing or decreasing. Finally, we use the derivative to findapproximate value of certain of Change of QuantitiesRecall that by the derivative dsdt, we mean the rate of change of distance s withrespect to the time t.

2 In a similar fashion, whenever one quantity y varies with anotherquantity x, satisfying some rule ( )yf x=, then dydx (or f (x)) represents the rate ofchange of y with respect to x and dydxx x =0 (or f (x0)) represents the rate of changeof y with respect to x at 0xx=.Further, if two variables x and y are varying with respect to another variable t, ,if ( )xf t=and ( )yg t=, then by Chain Ruledydx =dydxdtdt, if 0dxdt Chapter6 Application OFDERIVATIVES2021-22 Application OF DERIVATIVES195 Thus, the rate of change of y with respect to x can be calculated using the rate ofchange of y and that of x both with respect to us consider some 1 Find the rate of change of the area of a circle per second with respect toits radius r when r = 5 The area A of a circle with radius r is given by A = r2. Therefore, the rateof change of the area A with respect to its radius r is given by 2A() 2ddrrdrdr= = .When r = 5 cm, A10ddr= . Thus, the area of the circle is changing at the rate of10 cm2 2 The volume of a cube is increasing at a rate of 9 cubic centimetres persecond.

3 How fast is the surface area increasing when the length of an edge is 10centimetres ?Solution Let x be the length of a side, V be the volume and S be the surface area ofthe cube. Then, V = x3 and S = 6x2, where x is a function of time =9cm3/s (Given)Therefore9 =33V( )( )ddddxxxdtdtdxdt== (By Chain Rule)=23dxxdt ordxdt = (1)NowdSdt =22(6 )(6)dddxxxdtdxdt= (By Chain Rule)=233612xxx = (Using (1))Hence, whenx =10 cm, cm /sdSdt=2021-22 MATHEMATICS196 Example 3 A stone is dropped into a quiet lake and waves move in circles at a speedof 4cm per second. At the instant, when the radius of the circular wave is 10 cm, howfast is the enclosed area increasing?Solution The area A of a circle with radius r is given by A = r2. Therefore, the rateof change of area A with respect to time t isAddt =22()()dddrrrdtdrdt = = 2 r drdt(By Chain Rule)It is given thatdrdt =4cm/sTherefore, when r = 10 cm,Addt =2 (10) (4) = 80 Thus, the enclosed area is increasing at the rate of 80 cm2/s, when r = 10 dydx is positive if y increases as x increases and is negative if y decreasesas x 4 The length x of a rectangle is decreasing at the rate of 3 cm/minute andthe width y is increasing at the rate of 2cm/minute.

4 When x =10cm and y = 6cm, findthe rates of change of (a) the perimeter and (b) the area of the Since the length x is decreasing and the width y is increasing with respect totime, we have3 cm/mindxdt= and2 cm/mindydt=(a)The perimeter P of a rectangle is given byP =2(x + y)ThereforePddt =223 22dxdtdydt+ = += ()cm/min(b)The area A of the rectangle is given byA =x . yThereforeAddt =dxdyyxdtdt + = 3(6) + 10(2)(as x = 10 cm and y = 6 cm)= 2 cm2/min2021-22 Application OF DERIVATIVES197 Example 5 The total cost C(x) in Rupees, associated with the production of x units ofan item is given byC(x) = x3 x2 + 30x + 5000 Find the marginal cost when 3 units are produced, where by marginal cost wemean the instantaneous rate of change of total cost at any level of Since marginal cost is the rate of change of total cost with respect to theoutput, we haveMarginalcost (MC) = (3 ) (2 ) 30dCxxdx= +Whenx = 3, MC = (3 ) (3) 30 += + 30 = , the required marginal cost is ` (nearly).

5 Example 6 The total revenue in Rupees received from the sale of x units of a productis given by R(x) = 3x2 + 36x + 5. Find the marginal revenue, when x = 5, where bymarginal revenue we mean the rate of change of total revenue with respect to thenumber of items sold at an Since marginal revenue is the rate of change of total revenue with respect tothe number of units sold, we haveMarginal Revenue(MR) =R636dxdx=+Whenx =5, MR = 6(5) + 36 = 66 Hence, the required marginal revenue is ` the rate of change of the area of a circle with respect to its radius r when(a)r = 3 cm(b)r = 4 volume of a cube is increasing at the rate of 8 cm3/s. How fast is thesurface area increasing when the length of an edge is 12 cm? radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rateat which the area of the circle is increasing when the radius is 10 edge of a variable cube is increasing at the rate of 3 cm/s. How fast is thevolume of the cube increasing when the edge is 10 cm long?

6 Stone is dropped into a quiet lake and waves move in circles at the speed of5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast isthe enclosed area increasing?2021-22 radius of a circle is increasing at the rate of cm/s. What is the rate ofincrease of its circumference? length x of a rectangle is decreasing at the rate of 5 cm/minute and thewidth y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, findthe rates of change of (a) the perimeter, and (b) the area of the balloon, which always remains spherical on inflation, is being inflated by pumpingin 900 cubic centimetres of gas per second. Find the rate at which the radius ofthe balloon increases when the radius is 15 balloon, which always remains spherical has a variable radius. Find the rate atwhich its volume is increasing with the radius when the later is 10 ladder 5 m long is leaning against a wall. The bottom of the ladder is pulledalong the ground, away from the wall, at the rate of 2cm/s.

7 How fast is its heighton the wall decreasing when the foot of the ladder is 4 m away from the wall ? particle moves along the curve 6y = x3 +2. Find the points on the curve atwhich the y-coordinate is changing 8 times as fast as the radius of an air bubble is increasing at the rate of 12cm/s. At what rate is thevolume of the bubble increasing when the radius is 1 cm? balloon, which always remains spherical, has a variable diameter 3(21)2x+.Find the rate of change of its volume with respect to is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a coneon the ground in such a way that the height of the cone is always one-sixth of theradius of the base. How fast is the height of the sand cone increasing when theheight is 4 cm? total cost C(x) in Rupees associated with the production of x units of anitem is given byC(x) = + 15x + the marginal cost when 17 units are total revenue in Rupees received from the sale of x units of a product isgiven byR(x) = 13x2 + 26x + the marginal revenue when x = the correct answer for questions 17 and rate of change of the area of a circle with respect to its radius r at r = 6 cm is(A)10 (B)12 (C)8 (D)11 2021-22 Application OF total revenue in Rupees received from the sale of x units of a product isgiven byR(x) = 3x2 + 36x + 5.

8 The marginal revenue, when x = 15 is(A)116(B)96(C)90(D) Increasing and Decreasing FunctionsIn this section, we will use differentiation to find out whether a function is increasing ordecreasing or the function f given by f(x) = x2, x R. The graph of this function is aparabola as given in Fig consider the graph (Fig ) to the right of the origin. Observe that as wemove from left to right along the graph, the height of the graph continuously this reason, the function is said to be increasing for the real numbers x > consider the graph to the left of the origin and observe here that as we movefrom left to right along the graph, the height of the graph continuously , the function is said to be decreasing for the real numbers x < shall now give the following analytical definitions for a function which isincreasing or decreasing on an 1 Let I be an interval contained in the domain of a real valued function f is said to be(i)increasing on I if x1 < x2 in I f(x1) < f(x2) for all x1, x2 I.

9 (ii)decreasing on I, if x1, x2 in I f(x1) < f(x2) for all x1, x2 I.(iii)constant on I, if f(x) = c for all x I, where c is a (x) = x2 2432 94 1112 14 00 Values left to originas we move from left to right, theheight of the graph decreasesxf (x) = x2001214 113294 24 Values right to originas we move from left to right, theheight of the graph increases2021-22 MATHEMATICS200(iv)decreasing on I if x1 < x2 in I f(x1) f(x2) for all x1, x2 I.(v)strictly decreasing on I if x1 < x2 in I f(x1) > f(x2) for all x1, x2 graphical representation of such functions see Fig shall now define when a function is increasing or decreasing at a 2 Let x0 be a point in the domain of definition of a real valued function f is said to be increasing, decreasing at x0 if there exists an open interval Icontaining x0 such that f is increasing, decreasing, respectively, in us clarify this definition for the case of increasing 7 Show that the function given by f(x) = 7x 3 is increasing on Let x1 and x2 be any two numbers in R.

10 Thenx1 < x2 7x1 < 7x2 7x1 3 < 7x2 3 f(x1) < f(x2)Thus, by Definition 1, it follows that f is strictly increasing on shall now give the first derivative test for increasing and decreasing proof of this test requires the Mean Value Theorem studied in Chapter 1 Let f be continuous on [a, b] and differentiable on the open interval(a,b). Then(a)f is increasing in [a,b] if f (x) > 0 for each x (a, b)(b)f is decreasing in [a,b] if f (x) < 0 for each x (a, b)(c)f is a constant function in [a,b] if f (x) = 0 for each x (a, b)Strictly Increasing function(i)Neither Increasing norDecreasing function(iii)Strictly Decreasing function(ii)2021-22 Application OF DERIVATIVES201 Proof (a) Let x1, x2 [a, b] be such that x1 < , by Mean Value Theorem (Theorem 8 in Chapter 5), there exists a point cbetween x1 and x2 such thatf(x2) f(x1) = f (c) (x2 x1) (x2) f(x1) > 0(as f (c) > 0 (given)) (x2) >f(x1)Thus, we have121212( )( ), for all,[ , ]xxf xf xx xa b< Hence, f is an increasing function in [a,b].