Problem 16 - Calculus with Concepts in Calculus

Question

Find all critical numbers of the given function. $$ f(x)=1 /\left(e^{x}-1\right) $$

Step-by-Step Solution

Answer

The critical number is \( x = 0 \).

Step 1: Understand the Function

The function given is \( f(x) = \frac{1}{e^x - 1} \). Our goal is to find the critical numbers, where the derivative is either zero or undefined.

Step 2: Compute the First Derivative

Apply the quotient rule to find the derivative. Let \( u = 1 \) and \( v = e^x - 1 \). The quotient rule states \( f'(x) = \frac{v'u - vu'}{v^2} \), where \( u' = 0 \) and \( v' = e^x \). Substituting these, we get \( f'(x) = \frac{0 - 1 imes e^x}{(e^x - 1)^2} = \frac{-e^x}{(e^x - 1)^2} \).

Step 3: Set the Derivative Equal to Zero

To find critical points, set the derivative equal to zero: \( \frac{-e^x}{(e^x - 1)^2} = 0 \). Since \( -e^x eq 0 \), there are no values of \( x \) that make the derivative zero.

Step 4: Determine Where the Derivative is Undefined

The derivative \( f'(x) = \frac{-e^x}{(e^x - 1)^2} \) is undefined where the denominator \((e^x - 1)^2\) is zero. Set \( e^x - 1 = 0 \) which gives \( e^x = 1 \) or \( x = 0 \). Thus, \( x = 0 \) is a critical number.

Key Concepts

Derivative Calculations

Calculating derivatives is a crucial skill in calculus, especially when you're trying to find critical numbers of a function. A derivative helps us understand the rate at which a function changes. This involves finding the derivative of a function with respect to its variable.

To calculate the derivative efficiently, we often rely on rules like the power rule, product rule, quotient rule, and chain rule. In the context of our specific function, \( f(x) = \frac{1}{e^x - 1} \), one must compute its first derivative to identify critical points where the function's behavior changes. This is where changes such as peaks or troughs (local maxima or minima) occur. The first derivative tells us where the slope of the tangent to the function is zero (horizontal) or where it's undefined. These locations often mark critical numbers.

Critical numbers are where the derivative is zero or undefined, as these indicate potential maximum, minimum, or points of inflection on the graph. Understanding and computing derivatives correctly is key to identifying these characteristics of functions.

Quotient Rule in Calculus

The quotient rule in calculus is essential when you need to find the derivative of a ratio of two functions. It is especially useful when dealing with functions defined as a fraction, like \( f(x) = \frac{1}{e^x - 1} \).

The quotient rule states that if you have functions \( u(x) \) and \( v(x) \), then the derivative \( f'(x) \) is given by:\[ f'(x) = \frac{v'(x)u(x) - u'(x)v(x)}{(v(x))^2} \] This formula may seem complex, but it breaks down into simpler steps:

• Identify the function in the numerator \( u(x) \) and its derivative \( u'(x) \).
• Identify the function in the denominator \( v(x) \) and its derivative \( v'(x) \).
• Apply the quotient rule formula to compute \( f'(x) \).

When applied to our example function, the rule helps us derive \( f'(x) = \frac{-e^x}{(e^x - 1)^2} \). This step is crucial and aligns well with our ultimate goal of finding critical points, as it prepares us to assess where the function's slope might be zero or undefined.

Undefined Derivative

An undefined derivative occurs when division by zero happens in the derivative expression. For critical point analysis, identifying where a derivative is undefined is as vital as knowing where it equals zero.

In our example function, the derivative is \( f'(x) = \frac{-e^x}{(e^x - 1)^2} \). We observe that the derivative is undefined whenever the denominator \((e^x - 1)^2\) equals zero. Solving for this gives \( e^x - 1 = 0 \), resulting in \( e^x = 1 \) or \( x = 0 \).

Thus, \( x = 0 \) is a critical point because it makes the derivative undefined. Recognizing such points tells us where the function might have a sharp corner, discontinuity, or vertical tangent. These points become critical numbers to examine thoroughly when understanding a function's graph and identifying its peculiar behaviors.