Problem 49 - Calculus an Applied Approach

Question

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{x+1}{x}} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=\sqrt{\frac{x+1}{x}}\) is \(f'(x)=\frac{1}{2\sqrt{x(x+1)}}\). After plotting them, it's observed that the derivative never becomes zero thus the original function has no local minimum or maximum, it's always increasing or decreasing.

Step 1: Finding the Derivative

To find the derivative of \(f(x)=\sqrt{\frac{x+1}{x}}\), it is more appropriate to rewrite it in a form easier for differentiation. Thus the function can be rewritten as \(f(x)=((x+1)/x)^{1/2}\). The power rule, the chain rule and the quotient rule will be used to find the derivative. Using these rules, the derivative of \(f(x)\) is obtained as \(f'(x)=\frac{1}{2\sqrt{x(x+1)}}\).

Step 2: Graphing the Function and its Derivative

The function and its derivative are to be plotted in the same coordinate system. To do this, use a graphing tool is used that can plot functions. The function \(f(x)=\sqrt{\frac{x+1}{x}}\) and its derivative \(f'(x)=\frac{1}{2\sqrt{x(x+1)}}\) will provide two separate curves on the graph.

Step 3: Discussing the Function's behavior

The last step is to analyze the behavior of the function where its derivative is zero. Looking at the formula of the derivative, it will never equal zero as the numerator is always 1, a constant, while the denominator is a a multiplication of two variables. So there will be no point on our original function where the slope of the tangent line will be zero. This means the function doesn't have any local minimum or maximum, it means the function is always decreasing or increasing.

Key Concepts

Derivative of a Function

Understanding the concept of a derivative is crucial in mathematics, especially when analyzing the behavior of functions. In simple terms, a derivative represents the rate at which a function is changing at any given point. For the function in question, represented by
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Graphing Functions

Graphs are visual representations helping to understand functions better. By graphing the original function, ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline ewline

Behavior of Functions

Analyzing the behaviors of functions includes understanding where they increase, decrease, or remain constant. When looking at the derivative of the function
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