Last Date: 10th July 2012
By differentiation, we can figure out following characteristics of a graph.
1) Which points are inflection points (also known as stationary points and critical points)
2) At which points function has local minimum and local maximum.
3) Whether graph of a function is decreasing or increasing on an interval.
4) Whether graph of a function is concave upward or concave downward.
2) At which points function has local minimum and local maximum.
3) Whether graph of a function is decreasing or increasing on an interval.
4) Whether graph of a function is concave upward or concave downward.
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1) Discontinuous at a point It means that a function y = f(x) is not defined at some point. Usually such a function possesses vertical asymptotes at such point. If we take the example of a function y = 1/x. We knoe that when x=0, then we get zero in the denominator and division by zero is not allowed, therefore, graph of y = 1/x has discontinuity at x = 0. Also, it possesses a verticle asymptote x = 0.
2) Continuous but not differentiable at a point Usually such points are of closed shapes e.g. circle, ellipse, etc. etc. Such graphs are continuous at its left most and right most end on the x-axis, but are not differentiable at that point. y = Sqrt(1-x^2) is one such function which is continuous at x = 1 but not differentiable at x = 1.
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1) It means that a function y = f(x) is not defined at some point. Usually such a function possesses vertical asymptotes at such point. If we take the example of a function y = 1/x. We knoe that when x=0, then we get zero in the denominator and division by zero is not allowed, therefore, graph of y = 1/x has discontinuity at x = 0. Also, it possesses a verticle asymptote x = 0.
2) Continuous but not differentiable at a point Usually such points are of closed shapes e.g. circle, ellipse, etc. etc. Such graphs are continuous at its left most and right most end on the x-axis, but are not differentiable at that point. y = Sqrt(1-x^2) is one such function which is continuous at x = 1 but not differentiable at x = 1.
