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12: Tangents and Gradients © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

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Gradients and Tangents We need to be able to find these points using algebra e.g. Find the coordinates of the points on the curve where the gradient equals 4 Gradient of curve = gradient of tangent = 4 Points with a Given Gradient

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Gradients and Tangents Points with a Given Gradient Gradient is 4 e.g. Find the coordinates of the points on the curve where the gradient is 4 The gradient of the curve is given bySolution: Quadratic equation with no linear x - term

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Gradients and Tangents Points with a Given Gradient The points on with gradient 4

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Gradients and Tangents SUMMARY To find the point(s) on a curve with a given gradient: let equal the given gradient solve the resulting equation find the gradient function

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Gradients and Tangents Find the coordinates of the points on the curves with the gradients given where the gradient is -2 1. where the gradient is 3 2. Ans: (-3, -6) Ans: (-2, 2) and (4, -88) ( Watch out for the common factor in the quadratic equation ) Exercises

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Gradients and Tangents (-1, 3) x Solution: At x = 1 So, the equation of the tangent is Gradient = -5 (-1, 3) on line: The gradient of a curve at a point and the gradient of the tangent at that point are equal The equation of a tangent e.g. 1 Find the equation of the tangent at the point (-1, 3) on the curve with equation

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Gradients and Tangents An Alternative Notation The notation for a function of x can be used instead of y. When is used, instead of using for the gradient function, we write ( We read this as “ f dashed x ” ) e.g. This notation is helpful if we need to substitute for x.

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Gradients and Tangents Solution: To use we need to know y at the point as well as x and m So, the equation of the tangent is From (1), (2, 2) on the line e.g. 2 Find the equation of the tangent where x = 2 on the curve with equation where

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Gradients and Tangents if the y -value at the point is not given, substitute the x -value into the equation of the curve to find y SUMMARY To find the equation of the tangent at a point on the curve : find the gradient function substitute the x -value into to find the gradient of the tangent, m substitute for y, m and x into to find c

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Gradients and Tangents Exercises Ans: Find the equation of the tangent to the curve 1. at the point (2, -1) Find the equation of the tangent to the curve2. at the point x = -1, where

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Gradients and Tangents

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13: Stationary Points © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

13: Stationary Points © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules.

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