Problem 45 - Applied Calculus

Question

If \(f(t)=2 t^{3}-4 t^{2}+3 t-1\), find \(f^{\prime}(t)\) and \(f^{\prime \prime}(t)\).

Step-by-Step Solution

Answer

\(f'(t) = 6t^2 - 8t + 3\), \(f''(t) = 12t - 8\).

Step 1: Understand the Problem

We are given a function \(f(t) = 2t^3 - 4t^2 + 3t - 1\). The task is to find its first derivative \(f'(t)\) and its second derivative \(f''(t)\). Derivatives are a way to show the rate at which a function is changing at any given point.

Step 2: Find the First Derivative

To find the first derivative \(f'(t)\), apply the power rule of differentiation, which states \(\frac{d}{dt}[t^n] = n \cdot t^{n-1}\). Apply this to each term of the function.- The derivative of \(2t^3\) is \(6t^2\).- The derivative of \(-4t^2\) is \(-8t\).- The derivative of \(3t\) is \(3\).- The derivative of \(-1\) is \(0\).Combine these derivatives: \[ f'(t) = 6t^2 - 8t + 3 \]

Step 3: Find the Second Derivative

Next, calculate the second derivative \(f''(t)\) by differentiating the first derivative \(f'(t) = 6t^2 - 8t + 3\) again.- The derivative of \(6t^2\) is \(12t\).- The derivative of \(-8t\) is \(-8\).- The derivative of \(3\) is \(0\).Combine these derivatives:\[ f''(t) = 12t - 8 \]

Step 4: Verify the Results

Double-check each step to ensure all calculations follow the differentiation rules accurately. Make sure the application of the power rule to each term in the original and subsequent functions is consistent.

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus that simplifies taking derivatives. It makes it straightforward to deal with polynomial functions. Whenever you see a term like \(t^n\), applying the power rule allows you to find its derivative quickly.
The formula is: \(\frac{d}{dt}[t^n] = n \cdot t^{n-1}\). This means you multiply by the current power of \(t\) and then reduce the power by one.

• For example, for the term \(2t^3\), the derivative is \(6t^2\). The 3 comes down to multiply with 2, and the power reduces to 2.
• Similarly, \(-4t^2\) becomes \(-8t\), as you multiply by the power 2 and decrease the power by one.

This concept underpins the process of taking derivatives for any polynomial expression.

First Derivative

The first derivative, denoted as \(f'(t)\), represents the rate of change or slope of the original function \(f(t)\) at any given point. It tells us how the function is increasing or decreasing.
Calculating the first derivative involves applying rules like the power rule to each term of the function separately.

• For the function \(f(t) = 2t^3 - 4t^2 + 3t - 1\), its first derivative is \(f'(t) = 6t^2 - 8t + 3\).

This expression now shows us how the slope of \(f(t)\) changes with respect to \(t\), which can be crucial for understanding trends and identifying peaks or troughs in the function.

Second Derivative

To find deeper insights into the behavior of the function \(f(t)\), we compute its second derivative, \(f''(t)\). The second derivative indicates the rate of change of the first derivative, or in simpler terms, it shows the concavity of the function.

• When \(f''(t) > 0\), the function is concave up, often indicating a local minimum.
• When \(f''(t) < 0\), the function is concave down, suggesting a local maximum.

For the given function \(f(t)\), the second derivative is \(f''(t) = 12t - 8\). This result helps in understanding the direction and speed at which the first derivative \(f'(t)\) itself changes.

Differentiation Rules

Differentiation rules are the set of guidelines that make finding derivatives systematic and manageable. The power rule is just one of these essential tools. Other common rules include:

• Constant Rule: The derivative of a constant is always zero.
• Sum Rule: The derivative of a sum is the sum of their derivatives. This allows us to differentiate each term individually.
• Product and Quotient Rules: Useful for more complex expressions where functions are multiplied or divided.

Understanding and applying these rules ensure that you can tackle almost any differentiation problem with confidence. In the case of our function, the power rule and sum rule were used effectively to find both the first and second derivatives.