Problem 32 - Thomas Calculus

Question

Find the first and second derivatives of the functions in Exercises \(31-38\) \(s=\frac{t^{2}+5 t-1}{t^{2}}\)

Step-by-Step Solution

Answer

First derivative: \( s' = -5t^{-2} + 2t^{-3} \); Second derivative: \( s'' = 10t^{-3} - 6t^{-4} \).

Step 1: Identify the Function

We are given the function \( s = \frac{t^{2} + 5t - 1}{t^{2}} \). This is a rational function where both the numerator and the denominator are polynomials.

Step 2: Rewrite the Function

To differentiate the function easily, let's rewrite it by dividing the terms in the numerator by the denominator: \( s = \frac{t^{2}}{t^{2}} + \frac{5t}{t^{2}} - \frac{1}{t^{2}} = 1 + 5t^{-1} - t^{-2} \).

Step 3: Find the First Derivative

Differentiate each term separately using the power rule. The power rule states that for any term \( at^n \), the derivative is \( ant^{n-1} \).- The derivative of \( 1 \) is \( 0 \).- The derivative of \( 5t^{-1} \) is \(-5t^{-2} \).- The derivative of \( -t^{-2} \) is \( 2t^{-3} \).So, the first derivative \( s' = -5t^{-2} + 2t^{-3} \).

Step 4: Find the Second Derivative

Differentiate the first derivative to find the second derivative.- The derivative of \( -5t^{-2} \) is \( 10t^{-3} \).- The derivative of \( 2t^{-3} \) is \( -6t^{-4} \).Thus, the second derivative \( s'' = 10t^{-3} - 6t^{-4} \).

Key Concepts

Derivatives

Derivatives are fundamental concepts in calculus used to quantify how a function changes with respect to its input variables. In simple terms, the derivative of a function at a point measures the rate at which the function's value is changing at that point. It's like finding the instantaneous speed of a moving car by looking at how far it moves over an extremely small time. For the function given in the problem, we need to find its first and second derivatives:

• First Derivative: Represents the function's rate of change or slope at any point.
• Second Derivative: Tells us how the rate of change itself is changing, providing insights into the function's concavity and the presence of any inflection points.

Taking derivatives involves applying various rules and techniques, like the power rule (explained in a later section), to find these rates of change. Mastering derivatives allows you to analyze trends, optimize functions, and predict future outcomes.

Rational Functions

Rational functions are expressions that involve polynomials in their numerator and denominator. They have the form\[ f(x) = \frac{p(x)}{q(x)} \]where both \( p(x) \) and \( q(x) \) are polynomials. In this exercise, our rational function is \[ s = \frac{t^{2} + 5t - 1}{t^{2}} \]Rational functions are interesting because they can exhibit diverse characteristics, such as vertical/horizontal asymptotes and intercepts. Differentiating rational functions can be challenging due to their complexity. However, simplifying the function by rewriting the expressions in terms of negative exponents can often make the differentiation process more straightforward. This technique is powerful as it breaks down the function into simpler terms, allowing for easier application of basic derivative rules, like the power rule. Understanding how to manipulate and differentiate rational functions is crucial for analyzing mathematical models in science and engineering.

Power Rule

The power rule is a handy tool in calculus used to differentiate functions that are polynomials or can be expressed in a polynomial form. It states:\[ \text{If } f(t) = at^n, \text{ then } f'(t) = ant^{n-1} \]Using the power rule simplifies finding derivatives significantly. Instead of performing complex algebraic operations, you can systematically reduce the power of any term by simply multiplying by the term's current exponent and decreasing that exponent by one.In our exercise, once we rewrote the function \[ s = 1 + 5t^{-1} - t^{-2} \]we could apply the power rule directly to find both the first and second derivatives. Each term was treated individually:

• For \(1\): Since constants have no rate of change, the derivative is 0.
• For \(5t^{-1}\): Using the power rule, its derivative is \(-5t^{-2}\).
• For \(-t^{-2}\): The derivative becomes \(2t^{-3}\).

This process was repeated for the first derivative to find the second derivative, illustrating the ease with which the power rule can be used once a function is properly formatted.