Gradients and Tangents = 6 Solution: = 2 difference in the x -values difference in the y -values x x e.g. Find the gradient of the line joining the points with coordinates and

Gradients and Tangents The gradient of a straight line is given by We use this idea to get the gradient at a point on a curve This branch of Mathematics is called Calculus Gradients are important as they measure the rate of change of one variable with another. For the graphs in this section, the gradient measures how y changes with x

Gradients and Tangents Tangent at (2, 4) x The Gradient at a point on a Curve Definition: The gradient of a point on a curve equals the gradient of the tangent at that point. e.g The gradient of the tangent at (2, 4) is So, the gradient of the curve at (2, 4) is 4

Gradients and Tangents The gradient changes as we move along a curve e.g.

Gradients and Tangents

The Rule for Differentiation

Gradients and Tangents The Rule for Differentiation

Gradients and Tangents

Differentiation from first principles Gradient of AP = A xx + h f(x) f(x+h) P

Differentiation from first principles Gradient of AP = A xx + h f(x) f(x+h) P

Differentiation from first principles Gradient of AP = A xx + h f(x) f(x+h) P

Differentiation from first principles Gradient of AP = A xx + h f(x) f(x+h) P

Differentiation from first principles Gradient of AP = A xx + h f(x) f(x+h) P

Differentiation from first principles Gradient of tangent at A = A x f(x)

Differentiation from first principles f(x) = x 2

Differentiation from first principles f(x) = x 3

Generally with h is written as δx And f(x+ δx)-f(x) is written as δy

Generally

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

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. for all values of x

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 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