Problem 12 - University Calculus: Early Transcendentals

Question

Find the first and second derivatives. $$r=\frac{12}{\theta}-\frac{4}{\theta^{3}}+\frac{1}{\theta^{4}}$$

Step-by-Step Solution

Answer

First derivative: \( r' = -12\theta^{-2} + 12\theta^{-4} - 4\theta^{-5} \). Second derivative: \( r'' = 24\theta^{-3} - 48\theta^{-5} + 20\theta^{-6} \).

Step 1: Identify the Function

The function given is a rational function expressed as \( r = \frac{12}{\theta} - \frac{4}{\theta^3} + \frac{1}{\theta^4} \). Simplified, it can be rewritten as \( r = 12\theta^{-1} - 4\theta^{-3} + \theta^{-4} \).

Step 2: Find the First Derivative

Use the power rule for differentiation, \( \frac{d}{d\theta} [\theta^n] = n\theta^{n-1} \). Compute: - \( \frac{d}{d\theta} [12\theta^{-1}] = -12\theta^{-2} \) - \( \frac{d}{d\theta} [-4\theta^{-3}] = 12\theta^{-4} \) - \( \frac{d}{d\theta} [\theta^{-4}] = -4\theta^{-5} \) Combine these to get the first derivative: \( r' = -12\theta^{-2} + 12\theta^{-4} - 4\theta^{-5} \).

Step 3: Find the Second Derivative

Apply the power rule again to the first derivative: - \( \frac{d}{d\theta} [-12\theta^{-2}] = 24\theta^{-3} \) - \( \frac{d}{d\theta} [12\theta^{-4}] = -48\theta^{-5} \) - \( \frac{d}{d\theta} [-4\theta^{-5}] = 20\theta^{-6} \) Combine these to get the second derivative: \( r'' = 24\theta^{-3} - 48\theta^{-5} + 20\theta^{-6} \).

Key Concepts

rational functions

A rational function is defined as a function that can be expressed as the ratio of two polynomial functions, where the denominator is not equal to zero. These functions are interesting because they can have asymptotes and discontinuities depending on where the denominator is zero.
In this exercise, we have the function:

• \( r = \frac{12}{\theta} - \frac{4}{\theta^3} + \frac{1}{\theta^4} \)

This is a rational function because each term is a polynomial over another polynomial, specifically a constant divided by a power of \(\theta\). When you see rational functions, converting them into a more derivative-friendly format, like rewriting with negative exponents, can simplify finding derivatives.

• By rewriting, we have: \( r = 12\theta^{-1} - 4\theta^{-3} + \theta^{-4} \)

This representation allows easy differentiation using the power rule.

power rule for differentiation

The power rule is a central tool in calculus for differentiation. It gives a straightforward way to find the derivative of a function of the form \( \theta^n \). The rule states:

• If \( f(\theta) = \theta^n \), then the derivative \( f'(\theta) = n\theta^{n-1} \).

This rule applies to any real number \( n \), making it versatile and powerful.
In the given exercise, we apply the power rule to each term individually:

• \( \frac{d}{d\theta} [12\theta^{-1}] = -12\theta^{-2} \)
• \( \frac{d}{d\theta} [-4\theta^{-3}] = 12\theta^{-4} \)
• \( \frac{d}{d\theta} [\theta^{-4}] = -4\theta^{-5} \)

These derivatives are then combined to find the first derivative of the original function. The ease of applying the power rule makes differentiating polynomial and rational functions direct and efficient.

second derivative

The second derivative of a function is found by differentiating the first derivative. It plays a key role in understanding the concavity of the function and helps in locating points of inflection.
From the original problem, we have the first derivative:

• \( r' = -12\theta^{-2} + 12\theta^{-4} - 4\theta^{-5} \)

Now, applying the power rule to each term again will give us the second derivative:

• \( \frac{d}{d\theta} [-12\theta^{-2}] = 24\theta^{-3} \)
• \( \frac{d}{d\theta} [12\theta^{-4}] = -48\theta^{-5} \)
• \( \frac{d}{d\theta} [-4\theta^{-5}] = 20\theta^{-6} \)

By combining these results, the second derivative is:

• \( r'' = 24\theta^{-3} - 48\theta^{-5} + 20\theta^{-6} \)

The second derivative helps us explore the behavior of the function in greater detail, revealing how the rate of change is itself changing.