Tangents and Gradients

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

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

Points with a Given Gradient The points on with gradient 4

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

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

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

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

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

The graphs of the increasing functions and are

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

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

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.

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

( We read this as “ f dashed x ” ) 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 ” ) This notation is helpful if we need to substitute for x. e.g.

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

SUMMARY To find the equation of the tangent at a point on the curve : if the y-value at the point is not given, substitute the x -value into the equation of the curve to find y 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

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

(-1, 3) on line: The equation of a tangent 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

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