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Chapter 1: Problem 101

Use the fact that \(\sec x=\frac{1}{\cos x}\) to explain why the maximum domain of \(y=\sec x\) consists of all real numbers except odd integer multiples of \(\pi / 2\).

Short Answer

Expert verified
The domain of \(y=\sec x\) is all real numbers except odd multiples of \(\pi/2\) due to cosine being zero there.

Step by step solution

01

Understand the Relationship Between Secant and Cosine

The secant function, denoted as \(\sec x\), is defined as the reciprocal of the cosine function, which can be expressed as \(\sec x = \frac{1}{\cos x}\). This means that wherever cosine is zero, secant will be undefined because division by zero is not possible.
02

Identify Points Where Cosine is Zero

The cosine function, \(\cos x\), becomes zero at odd integer multiples of \(\frac{\pi}{2}\). Mathematically, these points can be expressed as \(x = (2n + 1) \frac{\pi}{2}\), where \(n\) is any integer.
03

Define the Domain of Secant Function

Since \(\sec x = \frac{1}{\cos x}\) is undefined where \(\cos x = 0\), we exclude these points from the domain of the secant function. Therefore, the maximum domain of \(y=\sec x\) includes all real numbers except those where \(x = (2n + 1) \frac{\pi}{2}\). This ensures no division by zero occurs.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Secant Function
The secant function is an intriguing and essential trigonometric function. It is denoted as \( \sec x \) and is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). The secant function is related to the hypotenuse over the adjacent side of a right triangle in terms of angles. But its importance extends beyond basic geometry.
The secant function is particularly useful because it helps in solving problems involving angles and distances. A key aspect of the secant function is that it is undefined wherever the cosine function is zero. This is because dividing by zero is mathematically inadmissible. So, understanding where \( \cos x = 0 \) gives us insight into the behavior and limitations of \( \sec x \). This relationship is foundational in grasping more complex trigonometric concepts.
Cosine Function
The cosine function, symbolized as \( \cos x \), is one of the primary trigonometric functions. In a right triangle, it represents the ratio of the length of the adjacent side to the hypotenuse. However, its relevance goes far beyond simple triangles.
Mathematically, the cosine function is periodic with a period of \(2\pi\). It means that its values repeat after every \(2\pi\) interval. Interestingly, the cosine function's value becomes zero at specific points. These points are crucial when we work with the secant function, as they help define where \( \sec x \) becomes undefined. Specifically, \( \cos x = 0 \) at odd integer multiples of \( \frac{\pi}{2} \), such as \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2} \), and so on.
Understanding these zero points allows us to determine where the secant function can't be evaluated.
Domain of a Function
The domain of a function is a critical concept in mathematics. It defines the set of all possible input values (or \( x \)-values) for which the function is defined. For the secant function \( y = \sec x \), the domain is a bit intricate due to its dependency on the cosine function.
Since \( \sec x = \frac{1}{\cos x} \), the secant function is undefined where \( \cos x = 0 \). Thus, to find the domain, we exclude these points. Cosine is zero at odd integer multiples of \( \frac{\pi}{2} \), i.e., \( x = (2n + 1) \frac{\pi}{2} \), where \( n \) is any integer. So, the domain of \( \sec x \) includes all real numbers except these specific values.
This exclusion ensures the function doesn't encounter division by zero, keeping the function valid and manageable.

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Most popular questions from this chapter

Michaelis-Menten Equation Enzymes serve as catalysts in many chemical reactions in living systems. The simplest such reactions transform a single substrate into a product with the help of an enzyme. The Michaelis-Menten equation describes the rate of such enzymatically controlled reactions. The equation, which gives the relationship between the initial rate of the reaction, \(v\), and the concentration of the substrate, \(s\), is $$ v(s)=\frac{v_{\max } s}{s+K} $$ where \(v_{\max }\) is the maximum rate at which the product may be formed and \(K\) is called the Michaelis-Menten constant. Note that this equation has the same form as the Monod growth function. Given some data on the reaction rate \(v\), for different substrate concentrations \(s\), we would like to infer the parameters \(K\) and (a) The graph of \(v\) against \(s\) is nonlinear, so it is hard to determine \(K\) and \(v_{\max }\) directly from a graph of the function \(v(s) .\) In the remaining parts of this question you will be guided to transform your plot into one in which the dependent variable depends linearly on the independent variable. First plot, using a graphing calculator, or by hand, \(v(s)\) for the following values of \(K\) and \(v_{\max }\) : $$ \left(K, v_{\max }\right)=(1,1), \quad(2,1), \quad(1,2) $$ (b) Show that the Michaelis-Menten equation can be written in the form $$ \frac{1}{v}=\frac{K}{v_{\max }} \frac{1}{s}+\frac{1}{v_{\max }} $$

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Use the indicated base to logarithmically transform each exponential relationship so that a linear relationship results. Then use the indicated base to graph each relationship either in log or semilog transformed coordinates so that a straight line results. $$ y=2^{x} ; \text { base } 2 $$

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