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Introduction to Differentiation Motion Graphs

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Travel Graph Describe what is happening at each stage of this travel graph. 1 2 3 4 5

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Travel Graph 1 2 3 4 5

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What is the average speed of the car in each section? 1 2 3 4 5 What is the link between the speed of the car and the graph?

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Projectile Motion

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What shape is the flight path of the basketball?

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Projectile Motion h(t) (m) t (s) Describe what is happening at each stage of this projectile motion graph. 1 2 3 4 5

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Projectile Motion h(t) (m) t (s) 4 51 2 3

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Projectile Motion h(t) (m) t (s) What is the instantaneous speed of the projectile at each point? 1 4 5 How can we calculate this? 2 3

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Calculating the Gradient of a Curve h(t) (m) t (s) Draw a normal to the curve then a tangent at that point. 1 3 4 5 2 normal tangent

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Calculating the Gradient of a Curve h(t) (m) t (s) Calculate the gradient of the tangent. 1 3 4 5 normal tangent

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Calculating the Gradient of a Curve h(t) (m) t (s) The Rate of Change of the graph is equal to the gradient at that point. tangent (0.5,10) (1.5,20) The gradient of the tangent equals the gradient of the curve at this point. The basketball is travelling at 10m/s at this point.

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Gradient Function Use your calculated values for the gradients to complete the following table. x01234 gradient 20100– 10– 20 Now plot these points on squared paper.

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Gradient Function You have plotted the value of the gradient at each point x on the curve for 0 ≤ x ≤ 4. This is called the Gradient Function Find the equation of this function.

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Using (0,20) and (2,0) we get

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Projectile Motion Use your knowledge of quadratic functions to obtain the equation of this function.

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Roots at x = 0 an x = 4 When x = 2, y = 20

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Equation of the gradient function: Equation of the projectile curve: Can you spot a link ? Here is another pair that might help you:

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Gradient Function, Secant & Tangent Click image to open Flash animation: Differentiation Animation

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Differentiation : the gradient of f(x) y = f(x) x y P (x, f(x)) Q (x + h, f(x + h)) x x + h h The gradient of PQ is given by:

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Differentiation : the derivative of f(x) y = f(x) x y P (x, f(x)) Q (x + h, f(x + h)) x x + h T The gradient of tangent at P is given by:

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