Calculus A new Maths module that picks up from Algebra and Graphs. You will learn how to find gradients of graphs numerically and algebraically. You will also learn how to find the area under the graph numerically and algebraically. You will also learn how to find the equation of a line given the gradient equation. If you’re really lucky you will be revolving graphs to form volumes!!

Gradients Aims: To recap finding the gradient, y=mx+c and using tangents to find gradients of curves. To be able to work out the gradient of curves numerically. To spot patterns on how to find the gradient algebraically.

Gradients between two points A B A = (1,5) B = (7,14) How far up? _____ How far along? ___ Gradient ___ How long is the line?

Gradients between two points A B A = (1,2) B = (5,18) How far up? _____ How far along? ___ Gradient ___ How long is the line?

Gradients between two points A B A = (2,3) B = (5,9) How far up? _____ How far along? ___ Gradient ___ How long is the line?

On axis labelled -10 to 10. Plot these points. Join the pairs A (2,2) to B (6,6) C (2,4) to D (5,10) E (-6,0) to F (-3,9) G (-8,-3) to H (-2,0) I (0,-2) to J (4,-10) K (4,-5) to L (6,-1) M (-6,-7) to N (0,-7) O (5,-8) to P (8,4) Find the gradient for each line

Equations of Straight Lines All straight lines can have an equation of the form y = mx + c m is the gradient of the linec is the y-intercept Find the gradient and y-intercept for y= 4.8x - 6.9

Fill the gaps! EquationGradienty-intercept y = 3x + 1 y = - 0.4x y = 6.2 y + x = 5.2

y = mx + c If an equation is not given in the form y = mx + c to find out useful information about the line we must rearrange the equation first. Re-arrange the following equations into the form y = mx + c and hence state the gradient and intercept with the y-axis 2x + y = 43x + y = -6 x + y = 72x + 2y = 6

Measuring gradients y step x step tangent Straight lines y y = mx + c c 0 x Curves y x 0 y step x step m = gradient = P

Estimating Gradients of Curves Draw a tangent line on the curve at the required point. Measure the rise and the run. Use these to calculate the gradient. Try to draw the tangent carefully and measure along the axes accurately. Complete the 4 worksheet questions.

Quiz Visit edpuzzle.com Create an account Join the Class using code dPzPG4 Answer the questions in the assignment.

Y= x

Age Number

(60,76000) (70,53500) (80,31000) (90,8500)

Calculus is an area of Mathematics which was first explored by Greek Mathematicians such as Archimedes. However, it is relatively recently that the way we use Calculus was developed. One such Mathematician was Sir Isaac Newton. His work help us make sense of the world about us-gravity, force and motion.

y = x 2 The gradient of y = x 2 at the green point is the same as the blue tangent line. But this is hard to calculate numerically. We are going to find the gradient of the orange chord line. This will give us an estimate for the gradient of the blue line. The closer together we make the pink and green dots the closer the gradient gets to the gradient of the blue line. Tangent Chord A B B B B B

Finding a gradient if we only know the equation of the curve. For example: Find the gradient of y=x 2 at x = 3

P(3, 9) y = x 2 Q 1 (4, 16) Gradient of PQ 1 Incremental changes = 7 Q 2 (3.5, 12.25) Gradient of PQ 2 = 6.5 Q 3 (3.25, ) Gradient of PQ 3 = 6.25 As Q  P gradient  6 See gradient demo

Find an estimate for the gradient of y= x 3 at x=1

Find an estimate for the gradient of y= x 3 at x=2

Find an estimate for the gradient of y= 3x 2 at x=3

Exercise Find an estimate for the gradient of each of these curves at the given points: a.y= x 3 at x = 3 b.y = 2x 3 at x = 4 c.y = 4x 2 at x = 2 d.y = x 2 + 2x +3 at x = 2 e.y = 2x 2 - 6x +2 at x = 5 f.Y = e x at x = 2 g.Y = 2ln x at x = 0.5

Generalising Equationy = x 2 y = x 3 y = x 4 y = x 5 y = x n Gradient Formula This is called differentiation! It is a method for finding the gradient algebraically. Either the notation dy / dx or f’(x) is used.

In general If y = x n dy / dx = nx n-1

A few questions to try Differentiate: a)y = x 5 b) y = x 9 c) f(x) = x 11 d) f(x) = x 15 e) y = x 100 f) y = x 999

Worksheet Practice Complete the worksheet questions to practice this skill. You need to be confident with this idea and that this method can be used to find the gradient. If there is more then one term you can just apply the process to each separately.

Quiz Write the numbers 1 to 10. This is to check your understanding. We will keep revisiting this idea to keep it fresh in your memory.