Problem 35 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of the function \(f\) by using the rules of differentiation. Let \(f(x)=2 x^{3}-4 x\). Find: a. \(f^{\prime}(-2)\) b. \(f^{\prime}(0)\) c. \(f^{\prime}(2)\)

Step-by-Step Solution

Answer

a. \(f^{\prime}(-2) = 20\) b. \(f^{\prime}(0) = -4\) c. \(f^{\prime}(2) = 20\)

Step 1: Finding the derivative of the function

To find the derivative of the function, we will use the power rule: \(\frac{d}{dx} x^n = n\cdot x^{n-1}\) for each term of the function. The given function is: $$f(x) = 2x^3 - 4x$$ Taking the derivative of the function: $$f'(x)=\frac{d}{dx}(2x^3) - \frac{d}{dx}(4x)$$ Use the power rule for each term: $$f'(x)= 6x^2 - 4$$ Now, we have the derivative of the function, \(f'(x) = 6x^2 - 4\). Next, we will evaluate the derivative at the specified points.

Step 2: Evaluating the derivative at \(x=-2\)

Plug in \(x=-2\) into the derivative function to find \(f'(-2)\): $$f'(-2) = 6(-2)^2 - 4$$ $$f'(-2) = 6(4) - 4$$ $$f'(-2) = 24 - 4$$ $$f'(-2) = 20$$ So, \(f'(-2) = 20\).

Step 3: Evaluating the derivative at \(x=0\)

Plug in \(x=0\) into the derivative function to find \(f'(0)\): $$f'(0) = 6(0)^2 - 4$$ $$f'(0) = 6(0) - 4$$ $$f'(0) = 0 - 4$$ $$f'(0) = -4$$ So, \(f'(0) = -4\).

Step 4: Evaluating the derivative at \(x=2\)

Plug in \(x=2\) into the derivative function to find \(f'(2)\): $$f'(2) = 6(2)^2 - 4$$ $$f'(2) = 6(4) - 4$$ $$f'(2) = 24 - 4$$ $$f'(2) = 20$$ So, \(f'(2) = 20\). To summarize: a. \(f^{\prime}(-2) = 20\) b. \(f^{\prime}(0) = -4\) c. \(f^{\prime}(2) = 20\)

Key Concepts

Understanding the Power Rule

The power rule is a fundamental tool in calculus used to differentiate functions of the form \( x^n \). It states that if you have a function \( f(x) = x^n \), its derivative \( f'(x) \) is \( n \cdot x^{n-1} \). This means:

• You bring down the power to the front as a coefficient.
• You then subtract one from the power you initially had.

For example, let's consider the function \( f(x) = 2x^3 \):
The derivative involves using the power rule on \( x^3 \). You bring down the 3 (making it \( 3 \cdot\)), and reduce the power by 1, resulting in \( 3x^{2} \). Multiply by the constant 2 to get the derivative of the first term: \( 6x^2 \).
This simple method can be applied to each term separately and combined for the final derivative.

Evaluating Derivatives at Specific Values

Once you have found the derivative of a function, you can calculate its value at any specific point. This is called derivative evaluation. Evaluating the derivative tells you the slope of the tangent line to the function at a particular point.
Here’s how you evaluate it:

• Replace \( x \) in the derivative by the given number.
• Calculate the resulting expression to find the slope.

For instance, using the derivative \( f'(x) = 6x^2 - 4 \) of the function \( f(x) = 2x^3 - 4x \), you can find:
- \( f'(-2) = 6(-2)^2 - 4 = 20 \), showing the slope at \( x=-2 \) is 20.
- \( f'(0) = 6(0)^2 - 4 = -4 \), indicating at \( x=0 \) the slope is -4.
- \( f'(2) = 6(2)^2 - 4 = 20 \), meaning at \( x=2 \) the slope is 20. These calculations help in understanding how fast the function is changing at those specified points.

Connecting Function Derivatives

Function derivatives are a way to understand how a function changes. They give you insights into the behavior and trends within the function’s graph.
When you differentiate a function, such as \( f(x) = 2x^3 - 4x \), the resulting expression \( f'(x) = 6x^2 - 4 \) tells you several important things:

• Rate of Change: It shows how steep the function is at any point \( x \).
• Critical Points: Where the derivative equals zero, it indicates potential maximum or minimum points, or points of inflection in the graph.
• Tangents: The derivative equals the slope of the tangent line to the function at any point \( x \).

Exploring these aspects helps in predicting and visualizing function behavior. Thus, derivatives play a crucial role in calculus and help in solving real-world problems involving rates of change.