Lecture 2.3 Derivative of Inverse Functions

Lecture 2.3 Derivative of Inverse Functions - Chapter 2...

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Chapter 2: Derivatives and Differentials Lecture 3: Derivatives of Inverse Functions By Prof. Phan Quoc Khanh
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Lecture Outline 1. Inverse Functions 2. Exponential and Logarithmic Functions 3. The Natural Logarithm and Exponential 4. Inverse Trigonometric Functions 5. Hyperbolic Functions 11/14/2014 Prof. Phan Quoc Khanh
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1. Inverse Functions Definition : A function f is called a one-to- one function if it never takes on the same value twice; that is ( ) ( ) for f x f y x y ¹ ¹ 11/14/2014 Prof. Phan Quoc Khanh
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One-to-One Functions Horizontal Line Test : A function is one-to-one if no horizontal line intersects its graph twice 11/14/2014 Prof. Phan Quoc Khanh One-to-one function NOT one-to-one function
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Inverse Functions Definition. Let f : A B be a one-to-one function with domain A and range B. Then its inverse function f –1 has domain B and range A and is defined by 11/14/2014 Prof. Phan Quoc Khanh 1 ( ) ( ) f y x f x y - = =
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Inverse Functions The letter x is traditionally used as the independent variable. So, when we concentrate on f –1 rather than on f , we usually reverse the roles of x and y and write Calcellation Equations: 1 ( ) ( ) f x y f y x - = = 1 1 ( ( )) for every in ( ( )) for every in f f x x x A f f x x x B - - = = Prof. Phan Quoc Khanh 11/14/2014
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How to Find Inverse Functions 1. Write y=f(x) 2. Solve this equation for x in terms of y (if possible) 3. To express f -1 as a function of x , interchange x and y . The resulting equation is y = f -1 (x) Prof. Phan Quoc Khanh 11/14/2014
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Example. Find the inverse of the function f ( x ) = x 3 + 2 Solution: 1. We first write y=x 3 + 2 2. Then, we solve this equation for x : 3. Finally, we interchange x and y : 3 1 3 2 2 ( ) x y x y f y - = - = - = 1 3 ( ) 2 y f x x - = = - Prof. Phan Quoc Khanh 11/14/2014
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Graphs of Inverse Functions Let a point (a, b) be on the graph of f • Then b=f(a) and so a=f -1 (b) This means that the point (b, a) is on the graph of f -1 The points (a, b) and (b, a) are symmetric about the line y=x The graph of f -1 is obtained by reflecting the graph of f about the line y=x . Prof. Phan Quoc Khanh 11/14/2014
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Graph of inverse function 11/14/2014 Prof. Phan Quoc Khanh a a b b
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Graphs of Inverse Functions Example : Find an explicit formula for f -1 and use it to graph f -1 and f , and the line y=x in the same coordinates 2 2 a) ( ) 1 2 / , 0 b) ( ) 2 , 0 f x x x f x x x x = - > = + > Prof. Phan Quoc Khanh 11/14/2014
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Derivatives of Inverse Functions 11/14/2014 Prof. Phan Quoc Khanh 1 1 1 1 ( ), where ( ) 1 1 1 '( ) '( ) '( ( )) 1 Thus, ( ) '( ( )) y f x x f y dy dy f y dx dx f y f f x d f x dx f f x - - - - = = = = = =
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2. Exponential Functions… Exponential functions have the form 11/14/2014 Prof. Phan Quoc Khanh ( ) , 0 is constant The domain is (- , ) and the range is (0, ) x f x a a = >  
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