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© Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Gradients

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© Nuffield Foundation Gradients 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? ©2011 Google – Map data

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

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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

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

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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

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Gradients of functions of the form y = x n 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 = x n ? Equation of curveGradient function y = x 2 2x2x y = x 3 3x23x2 y = x 4 4x34x3 y = x 5

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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 = x n ? Gradients

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

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

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

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

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