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

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

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

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

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

Gradients and Tangents Find the coordinates of the points on the curves with the gradients given where the gradient is 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

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

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.

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

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

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

Gradients and Tangents