Explanation Given: f ( x ) = sin ( x + a ) − sin x y = ± a Formula used: If f and g are integrable and g ( x ) ≤ f ( x ) for x in [ a , b ] then ∫ a b g ( x ) d x ≤ ∫ a b f ( x ) d x Calculation: ( a ) cos x ≤ 1 Using the comparison theorem , ∫ a b cos x d x ≤ ∫ a b 1 d x [ sin x ] a b ≤ ( b − a ) sin b − sin a ≤ b − a since , − cos x ≤ 1 using the comparison theorem , ∫ a b ( − cos x ) d x ≤ ∫ a b 1 d x − sin ( b ) + sin ( a ) ≤ b − a so , ± ( sin b − sin a ) ≤ b − a ⇒ | ...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Question

Chapter 5.4, Problem 56E

To determine

(a) We need to prove that $| \sin a - \sin b | \leq | a - b |$ .

(b) We need to show from the result of (a) that the graph of $f (x) = \sin (x + a) - \sin x$ lies between the horizontal lines $y = \pm a$ .

(c) We need to plot $y = f (x)$ and the lines $y = \pm a$ for $a = 0.5$ and $a = 0.2$

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Chapter 5.4, Problem 55E

Chapter 5.5, Problem 1PQ

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CALCULUS W/SAPLING ACCESS >IC<

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(a) We need to prove that | sin a − sin b | ≤ | a − b | . (b) We need to show from the result of (a) that the graph of f ( x ) = sin ( x + a ) − sin x lies between the horizontal lines y = ± a . (c) We need to plot y = f ( x ) and the lines y = ± a for a = 0.5 and a = 0.2

Solution Summary: The author illustrates how the graph of f and g are integrable and the lines y = a are plotted.

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(a) We need to prove that $| \sin a - \sin b | \leq | a - b |$ .

(b) We need to show from the result of (a) that the graph of $f (x) = \sin (x + a) - \sin x$ lies between the horizontal lines $y = \pm a$ .

(c) We need to plot $y = f (x)$ and the lines $y = \pm a$ for $a = 0.5$ and $a = 0.2$

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Chapter 5 Solutions

(a) We need to prove that | sin a − sin b | ≤ | a − b | . (b) We need to show from the result of (a) that the graph of f ( x ) = sin ( x + a ) − sin x lies between the horizontal lines y = ± a . (c) We need to plot y = f ( x ) and the lines y = ± a for a = 0.5 and a = 0.2

Solution Summary: The author illustrates how the graph of f and g are integrable and the lines y = a are plotted.

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(a) We need to prove that |sina−sinb|≤|a−b|. (b) We need to show from the result of (a) that the graph of f(x)=sin(x+a)−sinxlies between the horizontal lines y=±a. (c) We need to plot y=f(x)and the lines y=±afor a=0.5and a=0.2

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(a) We need to prove that $| \sin a - \sin b | \leq | a - b |$ .

(b) We need to show from the result of (a) that the graph of $f (x) = \sin (x + a) - \sin x$ lies between the horizontal lines $y = \pm a$ .

(c) We need to plot $y = f (x)$ and the lines $y = \pm a$ for $a = 0.5$ and $a = 0.2$