Problem 45 - Calculus Early Transcendentals

Question

$$ \begin{array}{l}{\text { (a) } f^{\prime \prime \prime}(2), \text { where } f(x)=3 x^{2}-2} \\ {\text { (b) }\left.\frac{d^{2} y}{d x^{2}}\right|_{x=1}, \text { where } y=6 x^{5}-4 x^{2}}\end{array} \text { (c) }\left.\frac{d^{4}}{d x^{4}}\left[x^{-3}\right]\right|_{x=1} $$

Step-by-Step Solution

Answer

(a) 0, (b) 112, (c) 360

Step 1: Differentiate the function for part (a)

Given function: \( f(x) = 3x^2 - 2 \)First, differentiate the function to find \( f'(x) \):\[ f'(x) = \frac{d}{dx}(3x^2 - 2) = 6x \]Next, differentiate again to find \( f''(x) \):\[ f''(x) = \frac{d}{dx}(6x) = 6 \]Finally, differentiate a third time to find \( f'''(x) \):\[ f'''(x) = \frac{d}{dx}(6) = 0 \]

Step 2: Evaluate the derivative at x=2 for (a)

The third derivative is \( f'''(x) = 0 \). Thus, \( f'''(2) = 0 \).

Step 3: Differentiate the function for part (b)

Given function: \( y = 6x^5 - 4x^2 \)First, differentiate the function to find \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = \frac{d}{dx}(6x^5 - 4x^2) = 30x^4 - 8x \]Next, differentiate again to find \( \frac{d^2y}{dx^2} \):\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(30x^4 - 8x) = 120x^3 - 8 \]

Step 4: Evaluate the second derivative at x=1 for (b)

Evaluate \( \frac{d^2y}{dx^2} \) at \( x=1 \):\[ \left.\frac{d^2y}{dx^2}\right|_{x=1} = 120(1)^3 - 8 = 120 - 8 = 112 \]

Step 5: Differentiate the function for part (c)

Given function: \( f(x) = x^{-3} \)First, differentiate to find \( \frac{d}{dx}(x^{-3}) \):\[ \frac{d}{dx}(x^{-3}) = -3x^{-4} \]Next, find the second derivative:\[ \frac{d^2}{dx^2}(x^{-3}) = 12x^{-5} \]Then, find the third derivative:\[ \frac{d^3}{dx^3}(x^{-3}) = -60x^{-6} \]Finally, find the fourth derivative:\[ \frac{d^4}{dx^4}(x^{-3}) = 360x^{-7} \]

Step 6: Evaluate the fourth derivative at x=1 for (c)

Evaluate \( \frac{d^4}{dx^4}(x^{-3}) \) at \( x=1 \):\[ \left.\frac{d^4}{dx^4}(x^{-3})\right|_{x=1} = 360(1)^{-7} = 360 \]

Key Concepts

Differentiation Techniques

Differentiation is a fundamental concept in calculus that helps us understand how functions change. In the context of higher-order derivatives, it becomes a powerful tool for analyzing functions' behaviors at multiple levels. To compute these derivatives, we apply differentiation techniques repeatedly until we achieve the desired order.

For example:

• When given a polynomial function like \( f(x) = 3x^2 - 2 \), applying the power rule allows us to compute the first derivative \( f'(x) = 6x \).
• Continuing differentiating, the second derivative becomes \( f''(x) = 6 \), and the third derivative \( f'''(x) = 0 \).

These steps showcase the systematic approach needed to find the derivative of higher orders. Mastering these techniques is crucial for tackling complex calculus problems.

Calculus Problems

In calculus, solving problems often involves finding derivatives of various functions. Calculus problems can range from simple polynomials to more complex functions, and each requires a good grasp of differentiation principles.

When given a function like \( y = 6x^5 - 4x^2 \), you use differentiation to find the first derivative, \( \frac{dy}{dx} = 30x^4 - 8x \). By differentiating again, you obtain the second derivative \( \frac{d^2y}{dx^2} = 120x^3 - 8 \).

With these derivatives, you can solve problems by evaluating expressions at specific points. For instance, in step 4 of our example, evaluating the second derivative at \( x = 1 \) led to \( 112 \). This value provides insights into the behavior of the function at that point, helping to solve the given calculus problem effectively.

Mathematics Education

In mathematics education, the focus is on equipping students with the tools and understanding needed to tackle diverse problems. Learning to find higher-order derivatives is an essential part of this education, as it builds upon basic differentiation concepts.

Understanding the process of differentiating consecutively, as seen in the function \( f(x) = x^{-3} \), underscores the applicability of calculus in real-world scenarios. Here, differentiating through four orders gives us coefficients each time: first \( -3x^{-4} \), second \( 12x^{-5} \), third \( -60x^{-6} \), and fourth \( 360x^{-7} \).

The ability to follow through these steps teaches students not only about mathematical computations but also about logical problem-solving skills. By focusing on these core skills, mathematics education prepares students for more intricate topics in advanced mathematics and practical applications.