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Introduction |
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Differential calculus can be considered as mathematics of motion, growth and change where there is a motion, growth, change. Whenever there is variable forces producing acceleration, differential calculus is the right mathematics to apply. Application of derivatives are used to represent and interpret the rate at which quantities change with respect to another variable. Most of the changes are considered in terms of independent variable time. But there is no restriction that the changes are considered with respect to time only. |
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Rate of Change of Quantity |
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If a quantity y varies with respect to another quantity 'x' satisfying some rule y = f(x), in other words if y is a function x, then 
represents the rate of change of y with respect to x |
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Tangents and Normals |
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Let (dy/dx)(x1,y1) = m be the slope of the tangent at (x1, y1) on the curve y=f(x). |
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1. Equation of the tangent at (x1,y1) is  |
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2. Equation of the normal at (x1,y1) is  |
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Increasing and Decreasing Functions |
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Let f be a function defined on an interval I and let x1 and x2 be any two points on I. |
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(i) f is said to be increasing in the interval I,
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(ii) f(x) is said to be decreasing function if for x1, x2  I
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Maxima and Minima |
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A function f(x) is said to have a local maximum at x = a, if $ is a neighbourhood I of 'a', such that
f(a)  f(x) for all x I |
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The number f(a) is called the local maximum of f(x). The point a is called the point of maximum. |
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A function f(x) is said to have a local minimum at x = a, if $ is a neighbourhood I of 'a', such that
f(a)  f(x) for all x I |
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Here, f(a) is called the local minimum of f(x). The point a is called the point of minimum. |
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Rolle's Theorem and Mean Value Theorem |
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Rolle's Theorem: Let f be a real valued function in [a,b] such that |
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 f is continuous in [a,b]. |
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 f is differentiable in (a,b). |
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Langrange's Mean Value Theorem |
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Let f be real valued function in [a,b] such that, |
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1. f is continuous in [a,b]. |
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2. f is differentiable in (a,b). |
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ApproxiMations by Differentials |
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Let y = f (x) be a differentiable function of x, errors in x and y are denoted by dx and dy, we have |
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\ Error in y = f ' (x) dx. |
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Curve Sketching |
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The graph of a given function gives a visual presentation of the behaviour of the function involving a study of symmetry, rise and fall, region of existence, passage through specific points etc. |
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The following points are helpful to trace the curve. |
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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. |
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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. |
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c) Symmetry about y = x:
If the equation of the curve remains unaltered if x and y are interchanged, then the curve is symmetrical about y = x. |
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Summary |
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1. If m = 0 the tangent at (x1, y1) is parallel to x-axis.
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2. If m1 = m2 the curves touch each other.
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3. Rolle's theorem
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4. Langrange's Mean Value theorem
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5. Maxima and Minima
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6. Generally the following points are examined for tracing the curves:
(i) Symmetry about x - axis
(ii) Symmetry about y - axis
(iii) Symmetry in opposite quadrants
(iv) Symmetry about the line y = x
(v) Passage through specific points
(vi) Points of intersection with the axes
(vii) Regions where the curve is increasing or decreasing
(viii) Region of existence.
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Conclusion |
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In this chapter we have learnt the application of derivatives to rate measure, also we have used the geometrical measurement of dy/dx to find the equations of the tangent and normal to a curve at any point on the curve, angle of intersection of the curves.
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Welcome! 
