Mathematics

Applications of Derivatives - 3 Session Applications of Derivatives - 3

Session Objectives Rolle’s Theorem Geometrical Meaning Lagrange’s Mean Value Theorem Approximation of Differentials Class Exercise

Rolle’s Theorem Let f(x) be a real function defined in the closed interval [a, b] such that f(x) is continuous in the closed interval [a, b] (ii) f(x) is differentiable in the open interval (a, b). (iii) f(a) = f(b) Then, there is a point c in the open interval (a, b), such that

Geometrical Meaning There will be at least one point with in [a, b] where tangent of the curve will be parallel to x-axis.

Example - 1 Verify Rolle’s theorem for the function f(x) = x2 – 8x + 12 on the interval [2, 6]. Solution : We have f(x) = x2 – 8x + 12 (1) Given function f(x) is polynomial function. \ f(x) is continuous on [2, 6] (2) f'(x) = 2x – 8 exists in (2, 6) \ f(x) is differentiable in (2, 6)

Solution (3) f(2) = 22 – 8 x 2 + 12 = 0 and f(6) = 62 – 8 x 6 + 12 = 0 \ f(2) = f(6) \ All the conditions of Rolle’s theorem is satisfied. \ Three exists some such that f'(c) = 0 Hence, Rolle’s theorem is verified.

Example - 2 Using Rolle’s theorem, find the points on the curve where the tangent is parallel to x-axis. Solution: (1) Being a polynomial function, the given function is continuous on [0, 4]. Function is differentiable in (0, 4).

Con. All conditions of Rolle’s theorem are satisfied. Required point is (2, –4)

Lagrange’s Mean Value Theorem Let f(x) be a function defined on [a, b] such that it is continuous on [a, b]. (ii) it is differentiable on (a, b). Then,there exists a real number such that

Geometrical Meaning

Geometrical Meaning From the triangle AFB, By Lagrange’s Mean Value theorem,  Slope of the chord AB = Slope of the tangent at

Example - 3 Verify Lagranges Mean Value theorem for the function f(x) = x2 – 3x + 2 on [-1, 2]. Solution : The function f(x) being a polynomial function is continuous in [-1, 2]. (2) f'(x) = 2x – 3 exists in (-1, 2) f(x) is differentiable in (-1, 2)

Solution So, there exists at least one such that Hence, Lagrange's mean value theorem is verified.

Example - 4 Using Lagrange’s mean value theorem, find the point on the curve , where tangent is parallel to the chord joining (1, –2) and (2, 1). Solution: (1) The function being a polynomial function is continuous on [1, 2]. Function is differentiable in (1, 2). such that tangent is parallel to chord joining (1, –2) and (2, 1)

Contd.

Approximation of Differentials As by the definition of Hence, for small increment in x, change in y will be

Example - 5 Using differentials, find the approximate value of Solution :

Contd. = 6 + 0.08 = 6.08

Example - 6 Using differentials, find the approximate value of Solution :

Contd.

Solution So, there exists at least one where tangent is parallel to chord joining (3, 0) and (5, 4). At x = 4 Required point is (4, 1)

Class Exercise - 5 Using differentials, find the approximate value of up to 3 places of decimals. Solution :

Solution

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