difference in the y-values e.g. Find the gradient of the line joining the points with coordinates and Solution: difference in the y-values x 7 - 1 = 6 x 3 - 1 = 2 difference in the x-values

This branch of Mathematics is called Calculus The gradient of a straight line is given by We use this idea to get the gradient at a point on a curve 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 This branch of Mathematics is called Calculus

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. Tangent at (2, 4) x 12 3 The gradient of the tangent at (2, 4) is So, the gradient of the curve at (2, 4) is 4

The gradient changes as we move along a curve

The Rule for Differentiation

The Rule for Differentiation

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

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

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

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

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

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

Differentiation from first principles f(x) = x2

Differentiation from first principles f(x) = x3

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

Generally

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 equals 4 Gradient of curve = gradient of tangent = 4 We need to be able to find these points using algebra

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: for all values of x

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 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