Problem 45 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the first and second derivatives of the given function. \(f(x)=2 x^{3}-3 x^{2}+1\)

Step-by-Step Solution

Answer

The first derivative of the given function is \(f'(x) = 6x^2 - 6x\), and the second derivative is \(f''(x) = 12x - 6\).

Step 1: Find the first derivative

To find the first derivative of the given function \(f(x) = 2x^3 - 3x^2 + 1\), we need to differentiate it with respect to \(x\). This can be done by applying the power rule, which states that \(\frac{d}{dx}(x^n) = nx^{n-1}\). Differentiating the function with respect to x: \(f'(x) = \frac{d}{dx}(2x^3) - \frac{d}{dx}(3x^2) + \frac{d}{dx}(1)\)

Step 2: Apply the power rule for all terms

We'll apply the power rule to each term in the expression for the first derivative: \(f'(x) = 2(3x^{3-1}) - 3(2x^{2-1}) + 0\)

Step 3: Simplify the first derivative

Now, we'll simplify the expression for the first derivative we obtained in the previous step: \(f'(x) = 6x^2 - 6x\) Now, we have found the first derivative, \(f'(x)\). Next, we'll find the second derivative.

Step 4: Find the second derivative

To find the second derivative, we need to differentiate the first derivative we found in the previous steps with respect to \(x\). We have the first derivative, \(f'(x) = 6x^2 - 6x\). We will differentiate it with respect to \(x\) to obtain the second derivative. \(f''(x) = \frac{d^2}{dx^2}(6x^2) - \frac{d^2}{dx^2}(6x)\)

Step 5: Apply the power rule for all terms

We'll apply the power rule to each term in the expression for the second derivative: \(f''(x) = 6(2x^{2-1}) - 6(1x^{1-1})\)

Step 6: Simplify the second derivative

Now, we'll simplify the expression for the second derivative we obtained in the previous step: \(f''(x) = 12x - 6\) So, the second derivative of the given function is \(f''(x) = 12x - 6\).

Key Concepts

Understanding the First Derivative

The first derivative of a function gives us valuable information about the rate of change of the function. When you calculate the first derivative, you are essentially determining how fast or slow the function is changing at any point along its curve. This information helps us understand increasing and decreasing behavior.
In terms of the given function, which is \(f(x) = 2x^3 - 3x^2 + 1\), the first derivative \(f'(x) = 6x^2 - 6x\) tells us how steep the curve is at any point \(x\).

• If \(f'(x) > 0\), the function is increasing at that point.
• If \(f'(x) < 0\), the function is decreasing at that point.
• If \(f'(x) = 0\), this could represent a local minimum or maximum point, or a point of inflection.

Working with the first derivative allows you to analyze the behavior of the function more deeply, particularly in terms of its slope and position along its graph.

Exploring the Second Derivative

The second derivative of a function provides insight into the concavity of the function's graph. It helps us understand how the rate of change itself is changing. Essentially,

• The second derivative \(f''(x)\) tells us whether the curve is concave up or concave down.

For the function given, \(f''(x) = 12x - 6\). From this, we can learn about the curvature:

• If \(f''(x) > 0\), the function is concave up at that point, resembling a cup shape, which suggests a local minimum if \(f'(x) = 0\).
• If \(f''(x) < 0\), the function is concave down, resembling a cap, which suggests a local maximum if \(f'(x) = 0\).
• If \(f''(x) = 0\), you may find a point of inflection where the graph changes concavity.

In practice, the second derivative is a powerful tool for examining the underlying shape and acceleration of the graph.

The Basics of Differentiation

Differentiation is a core concept in calculus that involves finding the derivative of a function. This process helps us determine how a function behaves. Differentiation uses rules like the power rule to compute derivatives efficiently. The function \(f(x) = 2x^3 - 3x^2 + 1\) is a polynomial, and such functions are straightforward to differentiate.
By differentiating, you essentially find a new function which describes the slope of \(f(x)\) at any given point \(x\). This is essential in various fields like physics and engineering, where knowing how things change over time or space is critical. Whether you're exploring the growth rate of a population or the velocity of an object, differentiation gives a precise analytical tool for understanding these changes.

Applying the Power Rule

The power rule is a fundamental technique in calculus used to differentiate functions of the form \(x^n\). The rule states that if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). This rule simplifies the differentiation of polynomial functions significantly.
For the class exercise, the power rule was applied to \(f(x) = 2x^3 - 3x^2 + 1\):

• The term \(2x^3\) becomes \(6x^2\) after applying the power rule.
• The term \(-3x^2\) becomes \(-6x\).
• The constant term \(+1\) results in \(0\), as constants drop out when differentiated.

The power rule allows you to handle each term of a polynomial individually. It makes finding both the first and second derivatives a straightforward process. Mastery of this rule opens up much of calculus, as many functions can be decomposed to pieces that are easy to handle with it.