7 - 1 = 6

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

2. 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 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

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

4. The gradient changes as we move along a curve e.g.

7. The Rule for Differentiation

8. The Rule for Differentiation

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

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

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

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

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

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

17. Differentiation from first principles f(x) = x2

18. Differentiation from first principles f(x) = x3

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

20. Generally

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

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

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

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

27. e.g.2 Show that is an increasing function. Solution: for all values of x

28. The graphs of the increasing functions and are and

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

30. 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 1. Show that is a decreasing function and sketch its graph.

31. 2. Show that is an increasing function Solution: . Completing the square: which is the sum of a square which is Solutions and a positive number. Hence y is an increasing function.

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