ERRORS AND APPROXIMATIONS ERRORS AND APPROXIMATIONS INTRODUCTION ERRORS

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ERRORS AND APPROXIMATIONS

ERRORS AND APPROXIMATIONS

ERRORS AND APPROXIMATIONS INTRODUCTION

ERRORS AND APPROXIMATIONS INTRODUCTION

ERRORS AND APPROXIMATIONS Ø The most distinguished Switzerland mathematicians Jakob Bernoulli (1654 -1705) and

ERRORS AND APPROXIMATIONS Ø The most distinguished Switzerland mathematicians Jakob Bernoulli (1654 -1705) and Johann Bernoulli (1667 – 1748) realized the surprising power of calculus and applied the tool to a great diversity of problems. Ø There are many applications of differentiation in science and engineering.

ERRORS AND APPROXIMATIONS Ø Differentiation is used in analysis of finance, economics, physics and

ERRORS AND APPROXIMATIONS Ø Differentiation is used in analysis of finance, economics, physics and natural sciences. Ø In this section, the approximate values of a function and errors in measuring the quantities can be studied. Infinitesimals: Ø If x is a quantity, x is change in x and it is a small quantity when compared to ‘x’ and x x, x x 2, x x 3, …… are small quantities in the decreasing order of magnitude.

ERRORS AND APPROXIMATIONS Ø Then these quantities are order 2, order 3, and so

ERRORS AND APPROXIMATIONS Ø Then these quantities are order 2, order 3, and so on. called infinitesimals of order 1, Approximations: - Ø Let y = f(x) be a function defined on an interval A and x A. Ø If x is any change in x, then let “ y” be the corresponding change in y. Ø Then y = f(x + x) – f(x).

ERRORS AND APPROXIMATIONS Differential: - Ø If y = f(x) be a function defined

ERRORS AND APPROXIMATIONS Differential: - Ø If y = f(x) be a function defined on an interval ‘A’. Ø It is denoted by dy or df. Note : - y dy.

ERRORS AND APPROXIMATIONS Errors: - Ø If y = f(x) be a function defined

ERRORS AND APPROXIMATIONS Errors: - Ø If y = f(x) be a function defined on an interval ‘A’ and x A. Ø Let x be any change in x and y be the corresponding change in y then (i) y is called error in y.

ERRORS AND APPROXIMATIONS Note : - To find the approximate value of a function

ERRORS AND APPROXIMATIONS Note : - To find the approximate value of a function y = f(x) we use f(x + x) f(x) + df. Ø If f(x) = I(x), the identify function in x,

ERRORS AND APPROXIMATIONS Note : - If y = f(x) is a homogeneous function

ERRORS AND APPROXIMATIONS Note : - If y = f(x) is a homogeneous function of degree n in x then the relative error in y is n times the relative error in x. Ø The following formula will be useful in solving the problems.

ERRORS AND APPROXIMATIONS Circle: - Ø If r is the radius, x is the

ERRORS AND APPROXIMATIONS Circle: - Ø If r is the radius, x is the diameter, p is the perimeter (circumference) and ‘A’ is the area of a circle then (i) x =2 r (ii) p =2 r (or) p = x

ERRORS AND APPROXIMATIONS Sector: - Ø If r is the radius, l is the

ERRORS AND APPROXIMATIONS Sector: - Ø If r is the radius, l is the length of the arc, ‘P’ is the perimeter and A is the area of a sector, then (i) l = r (ii) p =l + 2 r (or) p = r + 2 r = r( +2)

ERRORS AND APPROXIMATIONS Cube: - Ø If x is the side, s is the

ERRORS AND APPROXIMATIONS Cube: - Ø If x is the side, s is the surface area and v is the volume of a cube then (i) s = 6 x 2 (ii) v = x 3

ERRORS AND APPROXIMATIONS Sphere: - Ø If r is the radius, s is the

ERRORS AND APPROXIMATIONS Sphere: - Ø If r is the radius, s is the surface area and v is the volume of a sphere then (i) S = 4 r 2

ERRORS AND APPROXIMATIONS Cylinder: - Ø If r is the radius(of cross section), h

ERRORS AND APPROXIMATIONS Cylinder: - Ø If r is the radius(of cross section), h is the height, L is the lateral surface area, S is the total surface area and v is the volume of a (right circular) cylinder then (i) L = 2 rh (ii) S = 2 rh + 2 r 2 (iii) v = r 2 h

ERRORS AND APPROXIMATIONS Cone: Ø If r is the radius of base, h is

ERRORS AND APPROXIMATIONS Cone: Ø If r is the radius of base, h is the height, ‘l’ is the slant height, is the semi vertical angle, is the vertical angle, L is the lateral surface area, S is the total surface area, v be the volume of a (right circular) cone then (i) l 2 = r 2 + h 2 (iii) = 2

ERRORS AND APPROXIMATIONS Simple pendulum: -

ERRORS AND APPROXIMATIONS Simple pendulum: -

ERRORS AND APPROXIMATIONS Thank you…

ERRORS AND APPROXIMATIONS Thank you…