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4: The function © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

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"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Module C3

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The Gradient of growth Functions of this type, with a > 1, are called functions. We’ve already met the function e.g. Autograph demo

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The Gradient of We will now investigate the gradient of Notice first that as x increases, y increases e.g. x x x x x x Autograph demo

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The Gradient of gradient Notice first that as x increases, y increases... and the also increases e.g. x x x x x x The gradient function xx x x x x We will now investigate the gradient of looksIt the same as but...

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The Gradient of We will now investigate the gradient of e.g. x The gradient function x e.g. gradient Notice first that as x increases, y increases... and the also increases

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The Gradient of What do you think will happen if we repeat the process for ? Well, goes up more steeply than so we get a similar result but the gradient function is above the curve. Putting the 2 graphs on the same axes...

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The Gradient of It can be shown that So, suggesting that there is a value of a between 2 and 3 where the gradient of is equal to. The 1 st gradient graph is under the original curve... and the 2 nd is above the curve...

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The Gradient ofgradient of equals The value of a where the is an irrational number, written as e, where

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The Gradient of Using a letter for an irrational number isn’t a new idea to you. gradient of equals The value of a where the is an irrational number, written as e, where You have used ( the Greek p ) for

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The Gradient ofgradient of equals The value of a where the is an irrational number, written as e, where

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More Indices and Logs The function contains the index x, so x is a log. BUT the base of the log is e not 10, so ( since an index is a log ) We write as ( n for natural ) so, Logs with a base e are called natural logs We know that

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The Inverse of We can sketch the inverse by reflecting in y = x. is a one-to-one function so has an inverse function. Finding the equation of the inverse function is easy! So, N.B. The domain is. So Forwards x e it = f(x) Backwards opposite of e it is ln it Autograph demo

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SUMMARY is a growth function. (3 d.p.) At every point on, the gradient equals y : The inverse of is ( log with base e ) is defined for x > 0 only

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Can you suggest equations for the unlabelled graphs below? Both graphs are stretches of. HINT:

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This... is a stretch with scale factor 2 parallel to the y -axis. The equation is

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This... is a stretch with scale factor parallel to the x -axis. The equation is

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The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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More Indices and Logs The function contains the index x, so x is a log. BUT the base of the log is e not 10, so We know that ( since an index is a log ) We write as ( n for natural ) so, Logs with a base e are called natural logs

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SUMMARY is a growth function. (3 d.p.) At every point on, the gradient equals y : The inverse of is ( log with base e ) is defined for x > 0 only

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