Problem 3 - University Calculus: Early Transcendentals

Question

Find the first and second derivatives. $$s=5 t^{3}-3 t^{5}$$

Step-by-Step Solution

Answer

First derivative: \(15t^2 - 15t^4\), second derivative: \(30t - 60t^3\).

Step 1: Identify the Function

The given function is a polynomial expressed as \( s = 5t^3 - 3t^5 \). This needs to be differentiated to find both the first and second derivatives.

Step 2: First Derivative: Apply Power Rule

To find the first derivative \( \frac{ds}{dt} \), apply the power rule of differentiation \( \frac{d}{dt}[t^n] = nt^{n-1} \) to each term separately.- Differentiate \( 5t^3 \) to get \( 15t^2 \).- Differentiate \( -3t^5 \) to get \( -15t^4 \).Thus, the first derivative is \( \frac{ds}{dt} = 15t^2 - 15t^4 \).

Step 3: Second Derivative: Apply Power Rule Again

To find the second derivative \( \frac{d^2s}{dt^2} \), differentiate the first derivative \( 15t^2 - 15t^4 \) again using the power rule:- Differentiate \( 15t^2 \) to get \( 30t \).- Differentiate \( -15t^4 \) to get \( -60t^3 \).The second derivative is \( \frac{d^2s}{dt^2} = 30t - 60t^3 \).

Key Concepts

Derivatives

In calculus, a derivative represents the rate at which a function is changing at any given point. Imagine you're driving a car. The speedometer tells you how fast you're going at that very moment. That's similar to what a derivative does for a function. It gives us the 'instantaneous' rate of change.
For example, in the exercise, we're looking at the function \(s = 5t^3 - 3t^5\). By finding the derivative \(\frac{ds}{dt}\), we're determining how the function \(s\) changes as \(t\) changes. This is essential in understanding how variables are related and how they behave over time.

• First Derivative: Indicates the rate of change of the function.
• Second Derivative: Provides info about the function's curvature or concavity.

Derivatives are a foundational concept in calculus and provide a way to predict and understand real-world phenomena.

Power Rule

The power rule is a straightforward way to differentiate functions of the form \(t^n\). This rule states that if you have \(t^n\), its derivative is \(nt^{n-1}\). It's a simple yet powerful tool that allows you to quickly find derivatives of polynomial functions.
In our exercise, we apply the power rule to each term of the polynomial separately. For \(5t^3\), the power rule gives us \(15t^2\). Likewise, for \(-3t^5\), it gives \(-15t^4\). By systematically applying this rule, we can efficiently find the first derivative.

• Aids in quickly calculating derivatives.
• Works well with polynomial functions.

Using the power rule simplifies differentiation tasks, making it an indispensable tool in calculus.

Polynomial Differentiation

Polynomial differentiation involves finding the derivative of functions that are made up of several terms, each consisting of a variable raised to an exponent. These functions look like sums of terms such as \(5t^3 - 3t^5\). Differentiating a polynomial involves tackling each term one by one.
When differentiating polynomials, we apply the power rule to each component term; that's what we did here. This process not only gives us the first derivative but also allows us to find the second derivative by differentiating the result again.

• Breaks down complex expressions into manageable parts.
• Enables finding higher-order derivatives.

Understanding polynomial differentiation gives insight into the behavior of diverse mathematical functions and allows prediction of how these functions evolve.