Problem 5 - Calculus an Applied Approach

Question

find the second derivative of the function. $$ g(t)=\frac{1}{3} t^{3}-4 t^{2}+2 t $$

Step-by-Step Solution

Answer

The second derivative of the function \(g(t) = \frac{1}{3} t^{3} - 4 t^{2} + 2 t\), is \(g''(t) = 2t - 8\).

Step 1: First derivative

Differentiate the function \(g(t) = \frac{1}{3} t^{3} - 4 t^{2} + 2 t\). This results in \(g'(t) = t^{2} - 8 t + 2 \) using the power rule.

Step 2: Second derivative

Differentiate the derivative we just calculated. This gives us the second derivative, which is \(g''(t) = 2t - 8 \).

Key Concepts

First Derivative

The first derivative of a function represents the rate at which the function is changing with respect to its variable. It is essentially the slope of the function at any given point. In our exercise, we have the function \( g(t) = \frac{1}{3} t^{3} - 4 t^{2} + 2 t \). Finding the first derivative \( g'(t) \) gives us insight into how the values of \( g(t) \) increase or decrease as \( t \) changes.To find the first derivative, we differentiate the original function. Using techniques like the Power Rule makes this process systematic and straightforward. Understanding the first derivative is a critical skill as it lays the foundation for evaluating how a function behaves. Additionally, it tells us about the slope of the tangent lines to the graph of the function.

Power Rule

The Power Rule is one of the simplest and most frequently used rules in differentiation. It allows us to differentiate functions in the form of \( ax^n \), where \( a \) is a constant and \( n \) is a power.When applying the Power Rule, you take the exponent \( n \), multiply it by the coefficient \( a \), and then subtract one from the exponent. Mathematically, the power rule is represented as \( \frac{d}{dx}[x^n] = nx^{n-1} \). This rule significantly simplifies the differentiation of polynomial terms.

• For the term \( \frac{1}{3}t^3 \), the derivative is \( 1/3 \times 3t^{3-1} = t^2 \)
• For \( -4t^2 \), the derivative becomes \( -8t \)
• For \( 2t \), the derivative is \( 2 \)

Combining these results gives us the first derivative \( g'(t) = t^2 - 8t + 2 \). By mastering the Power Rule, students can efficiently tackle the differentiation of complex polynomials.

Differentiation Steps

Differentiation involves calculating the derivative, which is the backbone of calculus. It shows how a function changes at any point. The process is intuitive and follows specific steps to ensure accuracy. Here's a closer look at these steps using our example.Starting with the function \( g(t) = \frac{1}{3} t^{3} - 4 t^{2} + 2 t \):1. **Identify each term in the function.** Break it down into components like constant multiplication and powers of the variable.2. **Apply the Power Rule to each term.** As outlined, calculate the derivative of each term separately: - For \( \frac{1}{3} t^3 \), the derivative becomes \( t^2 \). - The derivative of \( -4t^2 \) is \( -8t \). - The derivative of \( 2t \) is a constant \( 2 \).3. **Combine the results.** Aggregate all the differentiated parts to form the first derivative \( g'(t) = t^2 - 8t + 2 \).Once you have the first derivative, you can find further derivatives, such as the second derivative. Just repeat the differentiation process on the first derivative:- For \( g'(t) = t^2 - 8t + 2 \), applying the Power Rule again gives you \( g''(t) = 2t - 8 \).Each step builds upon the previous ones, ensuring the complete understanding of how a function's behavior is analyzed mathematically.