S5_Unit_1_Outcome_3 - Higher Unit 1 Higher Outcome 3 Using...

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www.mathsrevision.com Higher Unit 1 Higher Unit 1 www.mathsrevision.com www.mathsrevision.com Finding the gradient for a polynomial Differentiating Brackets ( Type 1 ) Differentiating Harder Terms (Type 2) Differentiating with Leibniz Notation Equation of a Tangent Line ( Type 3 ) Increasing / Decreasing functions Max / Min and inflexion Points Curve Sketching Optimization Higher Outcome 3 Mind Map of Chapter Using differentiation (Application)
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www.mathsrevision.com On a straight line the gradient remains constant, however with curves  the gradient changes continually, and the gradient at any point is in  fact the same as the gradient of the tangent  at that point. The sides of the half-pipe are very  steep(S) but it is not very steep near  the base(B). B S Higher Outcome 3
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www.mathsrevision.com A Gradient of tangent = gradient of curve at A B Gradient of tangent = gradient of curve at B Higher Outcome 3
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www.mathsrevision.com Higher Outcome 3 For the function  y = f(x)   we do this by taking  the point (x, f(x))    and another “very close point”   ((x+h), f(x+h)). Then we find the gradient between the two.   (x, f(x)) ((x+h), f(x+h)) True gradient Approx gradient To find the gradient at any point on a  curve we need to modify the gradient  formula  2 1 2 1 - - y y m x x =
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www.mathsrevision.com The gradient is not exactly the same but is  quite close to the actual value     We can improve the approximation by making the value of  h  smaller This means the two points are closer together.   (x, f(x)) ((x+h), f(x+h)) True gradient Approx gradient Higher Outcome 3
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www.mathsrevision.com We can improve upon this approximation   by making the value of  h  even  smaller.   (x, f(x)) ((x+h), f(x+h)) True gradient Approx gradient So the points are even closer together. Higher Outcome 3
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www.mathsrevision.com Higher Outcome 3 Derivative We have seen that on curves the gradient changes continually and is  dependant on the position on the curve.  ie the x-value of the given point. The process of finding the gradient is called   DIFFERENTIATING  or    FINDING THE DERIVATIVE (Gradient) Differentiating Finding the  GRADIENT Finding the rate of  change
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www.mathsrevision.com If the formula/equation of the curve is given by f(x) Then the derivative is called  f '(x)     -    “f dash x” There is a simple way  of finding f '(x) from f(x).  f(x)             f '(x)   2x 2 4x  4x 2 8x  5x 10 50x 9 6x 7          42x 6  x 3           3x 2   x 5           5x 4  x 99           99x 98 Derivative Higher Outcome 3 Have guessed the  rule yet !
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This note was uploaded on 02/13/2012 for the course MAT 205 math 205 taught by Professor Google during the Spring '10 term at University of Phoenix.

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S5_Unit_1_Outcome_3 - Higher Unit 1 Higher Outcome 3 Using...

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