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

Gradients and Tangents Module C1 AQA Edexcel OCR MEI/OCR Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

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 The points on with gradient 4 Points with a Given Gradient

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 Increasing and Decreasing Functions An increasing function is one whose gradient is always greater than or equal to zero. for all values of x A decreasing function has a gradient that is always negative or zero. for all values of x

Gradients and Tangents e.g.1 Show that is an increasing function Solution: a positive number ( 3 )  a perfect square ( which is positive or zero for all values of x, and for all values of x is the sum of a positive number ( 4 ) so, is an increasing function

Gradients and Tangents Solution: e.g.2 Show that is an increasing function. To show that is never negative ( in spite of the negative term ), we need to complete the square. is an increasing function. for all values of x Since a square is always greater than or equal to zero,

Gradients and Tangents The graphs of the increasing functions and are and

Gradients and Tangents Exercises 2. Show that is an increasing function and sketch its graph. 1. Show that is a decreasing function and sketch its graph. Solutions are on the next 2 slides.

Gradients and Tangents 1. Show that is a decreasing function and sketch its graph. Solutions Solution:. This is the product of a square which is always and a negative number, so for all x. Hence is a decreasing function.

Gradients and Tangents Solutions 2. Show that is an increasing function and sketch its graph. Solution:. Completing the square: which is the sum of a square which is and a positive number. Hence y is an increasing function.

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

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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 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 Increasing and Decreasing Functions An increasing function is one whose gradient is always greater than or equal to zero. for all values of x A decreasing function has a gradient that is always negative or zero. for all values of x

Gradients and Tangents Solution: e.g. Show that is an increasing function. To show that is never negative ( in spite of the negative term ), we need to complete the square. is an increasing function. for all values of x Since a square is always greater than or equal to zero,

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  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 SUMMARY

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

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