Nuffield FreeStanding Mathematics Activity Gradients Nuffield Foundation 2011

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Nuffield Free-Standing Mathematics Activity Gradients © Nuffield Foundation 2011

Nuffield Free-Standing Mathematics Activity Gradients © Nuffield Foundation 2011

Gradients 5 6 7 4 8 3 1 2 © 2011 Google – Map

Gradients 5 6 7 4 8 3 1 2 © 2011 Google – Map data Walking the dog Kerry goes on a walk. Where is the gradient of Kerry’s walk positive? Where is it negative? Is there any part of the walk with a zero gradient? Where is the gradient steepest? © Nuffield Foundation 2011

Gradients Height of a child on a swing When is the gradient positive? negative?

Gradients Height of a child on a swing When is the gradient positive? negative? zero? What is happening then? This activity shows how to find accurate values for the gradients of curves.

Measuring gradients Straight lines y Curves y = mx + c y tangent P

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

Gradient of y = P (3, 9) x step 2 x y step x

Gradient of y = P (3, 9) x step 2 x y step x step gives an approximate value for the gradient y step It can be calculated more accurately

Incremental changes y = x 2 Gradient of PQ 1 (4, 16) =7 Q

Incremental changes y = x 2 Gradient of PQ 1 (4, 16) =7 Q 2 (3. 5, 12. 25) Q 3 (3. 25, 10. 5625) Gradient of PQ 2 P(3, 9) = 6. 5 As Q ® P Gradient of PQ 3 = 6. 25 gradient ® 6

Gradients of functions of the form y = xn Equation of curve Gradient function

Gradients of functions of the form y = xn Equation of curve Gradient function y = x 2 2 x y = x 3 3 x 2 y = x 4 4 x 3 y = x 5 Think about • What do you think is the gradient function for y = x 5? How can you prove it? • What about y = x 6? • Can you suggest an expression for the gradient of the general function y = xn ?

Gradients Reflect on your work • Describe the way in which the gradient of

Gradients Reflect on your work • Describe the way in which the gradient of a curve can be found using a spreadsheet. • What advantages does this have on drawing a tangent to a hand-drawn graph? • What is the gradient function of y = xn ?

Extension: Differentiation Gradient of PQ y = x 2 Q(x + dx, (x +

Extension: Differentiation Gradient of PQ y = x 2 Q(x + dx, (x + dx)2) P(x, x 2) As Q ® P dx ® 0 gradient ® 2 x

Rules of differentiation Function y = x 2 y= x 3 Derivative = 2

Rules of differentiation Function y = x 2 y= x 3 Derivative = 2 x = 3 x 2 y = x 4 = 4 x 3 y = x 5 = 5 x 4 y = mx =m y=c =0 General rules y = xn = nx n – 1 y = ax n = nax n – 1

General Rule for y = ax n = nax n – 1 Example y

General Rule for y = ax n = nax n – 1 Example y = 2 x 3 – 9 x 2 + 12 x + 1 = 6 x 2 – 18 x + 12 x 0 0. 5 y 1 5 1 1. 5 2 2. 5 6 5. 5 5 6 gradient 12 4. 5 0 – 1. 5 0 4. 5 y y = 2 x 3 – 9 x 2 + 12 x + 1 maximum minimum 0 x

Example y maximum y = 2 x 3 – 9 x 2 + 12

Example y maximum y = 2 x 3 – 9 x 2 + 12 x + 1 minimum 0 1 2 x Gradient function = 6 x 2 – 18 x + 12 0 1 2 x