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| Summary |
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| Let y = f(x) be a smooth curve and P(x,y) be a point on the curve. |
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 Equation of the tangent at (x1, y1) in the curve y = f (x) is |
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y - y1  |
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Equation of the normal at (x1, y1) in the curve y = f (x) is |
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If m = 0 the tangent at (x1, y1) is parallel to x-axis. |
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Angle of intersection of the curves is the acute angle between their tangents at the point of intersection. |
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| Let y = f1(x) and y = f2(x) be two curves intersecting at P. |
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If m1 = m2 the curves touch each other. |
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If m1m2 = -1 the curves are orthogonal. |
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The condition for the function f(x) to be increasing at x = a if |
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| f'(a) >0. |
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The condition for f(x) to be decreasing at x = a if f'(a) < 0. |
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| A function f(x) is said to be strictly increasing at x = a if f(x) > f(a) whenever x > a in the neighbour hood of a. |
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| A function f(x) is said to be strictly decreasing at x = a if f(x) < f(a) whenever x>a in the neighbour hood of a |
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If f'(a) = 0 then x = a is called a stationary point. |
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| If f'(a)=0, x=a is called the stationary point and the tangent at (a,f(a)) is parallel to x - axis. |
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A function f(x) is said to have a maximum at x = a if |
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The conditions for y = f(x) to have a maximum at x = a are |
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A function f(x) is said to have a minimum at x = a |
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The conditions for y = f(x) to have a minimum at x = a are |
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| x = a is called a point of inflexion. |
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Rolle's theorem: If a function f(x) is such that |
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| (i) f (x) is continuous on [a,b] |
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| (ii) f (x) is differentiable on (a,b) and |
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| (iii) f (a) = f (b) |
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| Geometrical interpretation of Rolle's theorem |
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| Let AB be the graph of y = f(x) such that A = (a,f(a)) and B = (b,f(b)) |
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| (i) f (x) is continuous between A and B. |
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| (ii) f (x) has derivative between A and B i.e., there is a unqiue tangent at every point between A and B. |
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| (iii) f (a) = f(b) |
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| at C is parallel to x-axis. |
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Langrange's Mean Value theorem: If a function f (x) is such that |
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| (i) f (x) is continuous on [a,b] and |
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| (ii) f (x) is differentiable on (a,b) then |
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| Geometrical interpretation of Lagrange's Mean value theorem. |
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| Let A(a, f(a)) and B(b, f(b)) be two points on the curve y = f (x) such that |
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| (i) f (x) is continuous between A and B. |
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| (ii) f (x) is differentiable between A and B |
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| i.e., there is a unique tangent at every point between A and B. |
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| then there exists a point C(c, f(c)) between A and B such that the tangent at C is parallel to the chord AB. |
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| Note that the point C is not unique. |
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| Relation between dy and dy |
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| Let A(x, y) and B(x + dx, y + dy) be two neighbouring points on the curve y = f(x). |
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| Let dx and dy be the differentiables of x and y respectively. |
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| AC = dx = dx |
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| BC = dy |
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| DC = dy |
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| dy = f'(x)dx |
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| dy - dy = BC - CD = BD |
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| \ The differential 'dy' and the increment 'dy' may not be equal. |
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Generally the following points are examined for tracing the curves |
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| (i) Symmetry about x - axis |
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| (ii) Symmetry about y - axis |
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| (iii) Symmetry in opposite quadrants |
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| (iv) Symmetry about the line y = x |
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| (v) Passage through specific points |
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| (vi) Points of intersection with the axes |
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| (vii) Regions where the curve is increasing or decreasing |
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| (viii) Region of existence. |
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