(b) Explanation Given: Given function ∫ a b f ( x ) g ( x ) d x = 0 . Proof: Consider functions f ( x ) and g ( x ) are orthogonal on [ a , b ] if ∫ a b f ( x ) g ( x ) d x = 0 . Consider below function, f ( x ) = x 2 , and g ( x ) = x 3 − 5 x To prove that: functions f ( x ) and g ( x ) are orthogonal on [ − b , b ] for any positive b . Since, function f and g are orthogonal on [ a , b ] if ∫ a b f ( x ) g ( x ) d x = 0 . Therefore, functions f ( x ) and g ( x ) will be orthogonal on [ − b , b ] if ∫ − b b f ( x ) g ( x ) d x = 0 (b) To determine To prove: that the sin k x and sin n x are orthogonal on [ − π , π ] .

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 9.3, Problem 53PS

(b)

To determine

To prove: that the two functions x2 and x3−5x are orthogonal on [−b,b] for any positive b .

(b)

To determine

To prove:that the sinkx and sinnx are orthogonal on [−π,π] .

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Chapter 9.3, Problem 52PS

Chapter 9.3, Problem 54PS

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Problem 1 Let f(x)=3. For this problem, we will use the notation cZ to denote the set of multiples of c. For example, 2Z denotes the even integers, and 3Z denotes multiples of 3. (a) Prove that f: Z → Z is not a function (that is, it is not a function if both its domain and codomain are integers.) (b) Prove that f: 2Z → Z is a function (that is, it is a function if its domain is even integers and its codomain is all integers.) Note: f is a function from A to B if and only if for each a € A, f(a) is defined, for each a € A, f(a) does not produce two different outputs, and • for each a € A, f(a) € B. Thus to prove f is a function you would show it has these 3 properties. To show that f is not a function you would show it does not have at least one of these 3 properties. See our first lecture on functions for examples and as reference on how to structure your proofs in this problem.

Chapter 9 Solutions

Calculus

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To prove: that the two functions x 2 and x 3 − 5 x are orthogonal on [ − b , b ] for any positive b .

Solution Summary: The author proves that the two functions x2 and