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**Nuffield Free-Standing Mathematics Activity**

Gradients

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**Gradients Walking the dog**

6 5 7 4 8 3 1 2 Walking the dog ©2011 Google – Map data 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?

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**Gradients When is the gradient positive? negative? zero?**

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.

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**Measuring gradients Straight lines Curves m = gradient = y y = mx + c**

x Curves y x tangent y step x step P y step x step m = gradient =

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**Gradient of y = x2 gives an approximate value for the gradient step x**

y step x step P (3, 9) It can be calculated more accurately

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**Incremental changes y = x2 Gradient of PQ1 = 7 Gradient of PQ2 = 6.5**

As Q ® P gradient ® 6 = 6.25

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**Gradients of functions of the form y = xn**

Equation of curve Gradient function y = x2 2x y = x3 3x2 y = x4 4x3 y = x5 Think about What do you think is the gradient function for y = x5? How can you prove it? What about y = x6? Can you suggest an expression for the gradient of the general function y = xn ?

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**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 ?

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**Extension: Differentiation**

P(x, x2) y = x2 Q(x + dx, (x + dx)2) Gradient of PQ As Q ® P dx ® 0 gradient ® 2x

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**Rules of differentiation**

Function Derivative General rules y = x2 = 2x y = x n = nx n – 1 = 3x2 y = x3 = 4x3 y = x4 y = ax n = 5x4 = nax n – 1 y = x5 = m y = mx = 0 y = c

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**General Rule for y = ax n Example y = 2x3 – 9x2 + 12x + 1 = 6x2 – 18x**

= nax n – 1 Example y = 2x3 – 9x2 + 12x + 1 = 6x2 – 18x + 12 0.5 1 1.5 2 2.5 x gradient y y x y = 2x3 – 9x2 + 12x + 1 12 1 4.5 5 maximum 6 – 1.5 5.5 minimum 5 4.5 6

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**Example y = 2x3 – 9x2 + 12x + 1 Gradient function = 6x2 – 18x + 12 y 1**

1 maximum 1 2 minimum = 6x2 – 18x + 12 Gradient function x

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TANGENCY Example 17Prove that the line 2x + y = 19 is a tangent to the circle x 2 + y 2 - 6x + 4y - 32 = 0, and also find the point of contact. ********

TANGENCY Example 17Prove that the line 2x + y = 19 is a tangent to the circle x 2 + y 2 - 6x + 4y - 32 = 0, and also find the point of contact. ********

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