Problem 8 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find the second derivative of the function. $$ g(t)=32 t^{-2} $$

Step-by-Step Solution

Answer

The second derivative of the function \(g(t)=32t^{-2}\) is \(g''(t)=192/t^4\).

Step 1: Finding the First Derivative

We have the function \(g(t)=32t^{-2}\), and we can rewrite it as \(g(t)=32/t^2\). Here, we apply power rule to find its first derivative. The power rule states that \((x^n)' = nx^{n-1}\). So the first derivative of the function g is obtained by multiplying the exponent (-2) by the coefficient (32) and decreasing the exponent by one: \(g'(t)=-2*32t^{-2-1}=-64t^{-3}=-64/t^3\).

Step 2: Finding the Second Derivative

Now, we compute the second derivative by differentiating the first derivative, once again using the power rule. Hence, \(g''(t)=-3*(-64)*t^{-3-1}=192t^{-4}=192/t^4\).

Step 3: Writing the Answer

The second derivative of the function \(g(t)=32t^{-2}\) is \(g''(t)=192/t^4\).

Key Concepts

Understanding the Power Rule

The power rule is a fundamental concept in calculus used to differentiate expressions where the variable is raised to a power. It simplifies the process of differentiation, making it quicker and more efficient. The rule can be simply stated as: if you have a function of the form \(x^n\), the derivative is \(nx^{n-1}\). In essence, the power (n) becomes a multiplier, and you reduce the power by one.

• This process helps in quickly finding derivatives without complex calculations.
• It's applicable when the function is a monomial term, like \(x^5\), \(t^{-3}\), etc.

For example, when differentiating \(32t^{-2}\), we multiply \(-2\) by \(32\) (resulting in \(-64\)) and subtract one from the exponent (resulting in \(-3\)). This power rule mechanism offers a graspable foundation for understanding how to derive functions quickly.

First Derivative: The Foundation

The concept of the first derivative serves as the stepping stone in the journey of differentiation. In calculus, the first derivative of a function provides the rate of change or "slope" at any given point on the curve of the function. By acquiring the first derivative, you gain insights into whether a function is increasing or decreasing, among other critical characteristics.
For the function \(g(t) = 32t^{-2}\), we applied the power rule to get the first derivative, both by reducing the exponent by one and multiplying by the original exponent. This led to \(g'(t) = -64t^{-3}\).

• First derivatives help in mapping out the function's behavior.
• You can interpret them to predict trends and directions of change.

Knowing how to calculate it effectively creates the foundation for other concepts like finding critical points and determining function concavity.

Exploring Differentiation

Differentiation is the core process in calculus that involves finding the derivative of a function. It's a powerful tool used to determine rates of change and analyze function behavior. Differentiation allows you to understand phenomena from physics, economics, and other scientific disciplines by quantifying how variables evolve over time.

• It's essentially about breaking down changes into manageable parts.
• With differentiation, complex variable relationships become clear and quantifiable.

By differentiating a function like \(g(t) = 32t^{-2}\), we first compute the first derivative to see how \(g(t)\) changes at any single point \(t\). Further differentiation (like a second derivative) informs us regarding the acceleration of this change or how the rate itself is changing.
Tangibly grasping differentiation sets the stage for both theoretical and practical applications across varied fields.