DIFFERENTIATION APPLICATIONS 1

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Presentation transcript:

DIFFERENTIATION APPLICATIONS 1 PROGRAMME 8 DIFFERENTIATION APPLICATIONS 1

Programme 8: Differentiation applications 1 Equation of a straight line Tangents and normals to a curve at a given point Curvature Centre of curvature

Programme 8: Differentiation applications 1 Equation of a straight line Tangents and normals to a curve at a given point Curvature Centre of curvature

Programme 8: Differentiation applications 1 Equation of a straight line The basic equation of a straight line is: where:

Programme 8: Differentiation applications 1 Equation of a straight line Given the gradient m of a straight line and one point (x1, y1) through which it passes, the equation can be used in the form:

Programme 8: Differentiation applications 1 Equation of a straight line If the gradient of a straight line is m and the gradient of a second straight line is m1 where the two lines are mutually perpendicular then:

Programme 8: Differentiation applications 1 Equation of a straight line Tangents and normals to a curve at a given point Curvature Centre of curvature

Programme 8: Differentiation applications 1 Tangents and normals to a curve at a given point Tangent The gradient of a curve, y = f (x), at a point P with coordinates (x1, y1) is given by the derivative of y (the gradient of the tangent) at the point: The equation of the tangent can then be found from the equation:

Programme 8: Differentiation applications 1 Tangents and normals to a curve at a given point Normal The gradient of a curve, y = f (x), at a point P with coordinates (x1, y1) is given by the derivative of y (the gradient of the tangent) at the point: The equation of the normal (perpendicular to the tangent) can then be found from the equation:

Programme 8: Differentiation applications 1 Equation of a straight line Tangents and normals to a curve at a given point Curvature Centre of curvature

Programme 8: Differentiation applications 1 Curvature The curvature of a curve at a point on the curve is concerned with how quickly the curve is changing direction in the neighbourhood of that point. Given the gradients of the curve at two adjacent points P and Q it is possible to calculate the change in direction  = 2 - 1

Programme 8: Differentiation applications 1 Curvature The distance over which this change takes place is given by the arc PQ. For small changes in direction  and small arc distance s the average rate of change of direction with respect to arc length is then:

Programme 8: Differentiation applications 1 Curvature A small arc PQ approximates to the arc of a circle of radius R where: So and in the limit as Which is the curvature at P; R being the radius of curvature

Programme 8: Differentiation applications 1 Curvature The radius of curvature R can be shown to be given by:

Programme 8: Differentiation applications 1 Equation of a straight line Tangents and normals to a curve at a given point Curvature Centre of curvature

Programme 8: Differentiation applications 1 Centre of curvature If the centre of curvature C is located at the point (h, k) then:

Programme 8: Differentiation applications 1 Learning outcomes Evaluate the gradient of a straight line Recognize the relationship satisfied by two mutually perpendicular lines Derive the equations of a tangent and a normal to a curve Evaluate the curvature and radius of curvature at a point on a curve Locate the centre of curvature for a point on a curve