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Published byHarry Cooter Modified about 1 year ago

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C1: Tangents and Normals Learning Objective: to find the equation of a tangent and a normal to a curve at a given point by applying the rules of differentiation

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Starter: Differentiate 1.y = 2x 4 + 7x 3 - 5x 2 2.y = ½ x 4 3.y = 2√x 4.y = 4x x -2 5.y = (x 2 + 4)(x 2 -3) 6.y = (2x 8 - 3x 4 +5x 2 )/ x 3

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Tangents and normals Remember, the tangent to a curve at a point is a straight line that just touches the curve at that point. The normal to a curve at a point is a straight line that is perpendicular to the tangent at that point. We can use differentiation to find the equation of the tangent or the normal to a curve at a given point. For example: Find the equation of the tangent and the normal to the curve y = x 2 – 5 x + 8 at the point P(3, 2).

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Tangents and normals y = x 2 – 5 x + 8 At the point P(3, 2) x = 3 so: 4 The gradient of the tangent at P is therefore 4. Using y – y 1 = m ( x – x 1 ) give the equation of the tangent at the point P(3, 2): y – 2 = 4( x – 3) y – 2 = 4 x – 12 y – 4 x + 10 = 0

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Tangents and normals The normal to the curve at the point P(3, 2) is perpendicular to the tangent at that point. Using y – y 1 = m ( x – x 1 ) give the equation of the tangent at the point P(3, 2): The gradient of the tangent at P is 4 and so the gradient of the normal is

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Task 1 1.Find the equation of the tangent to the curve y = x 2 - 7x + 10 at the point (2, 0). 2.Find the equation of the normal to the curve y = x 2 – 5x at the point (6, 6). 3.Find the equations of the normals to the curve y = x + x 3 at the points (0, 0) and (1, 2), and find the co- ordinates of the point where these normals meet. 4.For f(x) = 12 – 4x + 2x 2, find an equation of the tangent and normal, at the point where x = -1 on the curve with equation y = f(x).

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