Consider a rational function \(f(x) = \frac{ax+b}{cx+d}\) representing a simplified model of a control system’s feedback mechanism, where \(a\), \(b\), \(c\), and \(d\) are constants.
(a) Set up the model by finding the conditions on the constants such that the function has a vertical asymptote at \(x=k\) and a horizontal asymptote at \(y=m\).
(b) Analyze the behavior of \(f(x)\) near the vertical asymptote \(x=k\).
(c) Discuss how the derivative \(f'(x)\) can be used to determine the rate of change of \(f(x)\) as \(x\) approaches the horizontal asymptote \(y=m\).
Question
Set up the model by finding the conditions on the constants \(a\), \(b\), \(c\), and \(d\) such that \(f(x)=\frac{ax+b}{cx+d}\) has a vertical asymptote at \(x=k\) and a horizontal asymptote at \(y=m\).
Hint
Hint
Hint
Consider where the denominator equals zero for the vertical asymptote and the ratio of the leading coefficients for the horizontal asymptote.
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Answer
Answer
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Explanation
Explanation
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Complete Answer
Vertical asymptote: Set \(cx+d=0\) at \(x=k\) so \(ck+d=0\) which gives \(d=-ck\). Horizontal asymptote: Since degrees are equal, \(y=\frac{a}{c}=m\) so \(a=cm\). The constant \(b\) is unrestricted.
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Answer Explanation
Answer Explanation
Question
Analyze the behavior of \(f(x)\) as \(x\) approaches the vertical asymptote \(x=k\).
Hint
Hint
Hint
Discuss the limits from the left and right of \(x=k\) and consider the sign of \(c\) in the expression \(f(x)=\frac{ax+b}{c(x-k)}\).
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Answer
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Explanation
Explanation
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Complete Answer
With \(d=-ck\), the function becomes \(f(x)=\frac{ax+b}{c(x-k)}\). As \(x\to k^-\) the denominator is negative (if \(c>0\)) causing \(f(x)\) to approach \(-\infty\), while as \(x\to k^+\) it is positive leading \(f(x)\) to approach \(\infty\). The reverse occurs if \(c<0\).
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Answer Explanation
Answer Explanation
Question
Discuss how the derivative \(f'(x)\) can be used to determine the rate of change of \(f(x)\) as \(x\) approaches the horizontal asymptote \(y=m\).
Hint
Hint
Hint
Differentiate \(f(x)\) using the quotient rule and examine its behavior as \(x\to\infty\) to see how it approaches 0.
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Answer
Answer
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Explanation
Explanation
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Complete Answer
Differentiating \(f(x)=\frac{ax+b}{cx+d}\) gives \(f'(x)=\frac{a(cx+d)-c(ax+b)}{(cx+d)^2}\), which simplifies to \(\frac{ad-cb}{(cx+d)^2}\). As \(x\to\infty\), \(cx+d\) becomes large, ensuring that \(f'(x)\to 0\); this indicates that the rate of change diminishes as \(f(x)\) approaches the horizontal asymptote \(y=m\).
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Answer Explanation
Answer Explanation
