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Let y = f (x) be a differentiable function of x, errors in x and y are denoted by d x and d y, we have \ Error in y = f ' (x) d ..

dx and dy are called the differentials of x and y respectivel..

Let y = f (x) = x 1 / 4 Let x = 81, d x =1. Taking these values we have ..

a) Symmetry about x - axis If the equation of the curve remains unaltered when y is replaced by -y, then the curve is symmetrical about x-axis. b) Symmetry about y-axis If the equation of the curve remains unaltered when x is replaced by -x then the curve is symmetrical about y-axis. c) Symmetry a..

Symmetry (a) By replacing y by -y, the equation (1) is altered, therefore the curve is not symmetrical about x-axis. (b) By replacing x by -x, the equation of the curve is altered, therefore the curve is not symmetrical about y-axis. (c) Replace x and y by -x and -y respectively in the equation y ..

The points of intersection with x-axis is determined by letting y = 0. Putting y = 0, - sin 2x = 0 This implies the curve intersects the x-axis at the points where The points at which the tangent is parallel to x-axes are determined by solving. y = - sin 2x The tangent is paral..

This section explains how derivative can be used to check whether a function is increasing, decreasing or neither increasing nor decreasing in its domain. Let f be a function defined on an interval I and let x 1 and x 2 be any two points on I. (i) f is said to be increasing in the interval ..

Let f be continuous on [a, b] and differentiable on the open interval (a, b). Then (a) f is increasing on [a, b] if f '(x) > 0 for each x (a, b) (b) f is decreasing on [a, b] if f '(x) < 0 for each x (a, b) This theorem can be proved by using Mean Value Theorem. We shall prove the theore..

Find the intervals on which the function (a) increasing (b) decreasing Differentiating the function, we have The critical values in ascending order are -1, 1. We divide the Real numbers into the intervals = - ve Since f '(x) < 0, the function is decreasing in the interval ..