Problem 148 - Calculus Volume 1

Question

A book publisher has a cost function given by \(C(x)=\frac{x^{3}+2 x+3}{x^{2}},\) where \(x\) is the number of copies of a book in thousands and \(C\) is the cost, per book, measured in dollars. Evaluate \(C^{\prime}(2)\) and explain its meaning.

Step-by-Step Solution

Answer

C'(2) = -0.25, meaning the cost decreases by $0.25 per book for 2000 copies.

Step 1: Understand the Cost Function

The given cost function is \(C(x)=\frac{x^{3}+2x+3}{x^{2}}\). This function describes the cost per book in dollars when producing \(x\) thousands of copies.

Step 2: Derive the Function Using Quotient Rule

To find the derivative \(C'(x)\), we apply the quotient rule. The quotient rule states that if we have a function \(\frac{u(x)}{v(x)}\), its derivative is \(\frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}\), where \(u(x) = x^3 + 2x + 3\) and \(v(x) = x^2\).

Step 3: Calculate Derivatives of the Numerator and Denominator

Compute \(u'(x)\): The derivative of \(x^3 + 2x + 3\) is \(3x^2 + 2\). Compute \(v'(x)\): The derivative of \(x^2\) is \(2x\).

Step 4: Apply Quotient Rule

Substitute into the quotient rule formula:\[C'(x) = \frac{x^2(3x^2 + 2) - (x^3 + 2x + 3)(2x)}{(x^2)^2}\]

Step 5: Simplify the Expression

Simplify \(C'(x)\):- \(x^2(3x^2 + 2) = 3x^4 + 2x^2\)- \((x^3 + 2x + 3)(2x) = 2x^4 + 4x^2 + 6x\)Substitute and simplify:\[C'(x) = \frac{3x^4 + 2x^2 - (2x^4 + 4x^2 + 6x)}{x^4} = \frac{x^4 - 2x^2 - 6x}{x^4}\]

Step 6: Evaluate the Derivative at x = 2

Plug \(x = 2\) into \(C'(x)\): \[C'(2) = \frac{2^4 - 2 \times 4 - 6 \times 2}{2^4} = \frac{16 - 8 - 12}{16} = \frac{-4}{16} = -0.25\]

Step 7: Interpret the Result

The value \(C'(2) = -0.25\) means that for small increases in production when producing 2000 copies, the cost per book decreases by $0.25 per additional book produced.

Key Concepts

Understanding Cost Function

A cost function is a crucial part of production and business analysis in calculus. In our exercise, the cost function, given by\[ C(x) = \frac{x^{3} + 2x + 3}{x^{2}} \]is used to determine the cost per book when producing \(x\) thousands of copies. This mathematical expression allows businesses to evaluate and predict production costs based on the number of items, which in this case is books.

• The numerator \(x^3 + 2x + 3\) represents the total cost related to producing \(x\) thousands of books.
• The denominator \(x^2\) adjusts that total into a per-book cost.

By understanding this function, businesses can identify their cost structure and make informed financial decisions about scaling production.

Grasping the Derivative

In calculus, a derivative represents how a function changes as its input changes. It's essentially the rate at which something is happening.
In this exercise, we need to find the derivative of the cost function, \(C(x)\). The process of finding a derivative is called differentiation, and it helps us determine the slope of a function at any given point. This slope tells us how a small change in the number of books produced affects the cost per book.
Here, understanding the derivative \(C'(x)\) is key to knowing how costs adjust as production varies.

Applying the Quotient Rule

Differentiating a fraction-like function requires the quotient rule, an essential concept in calculus. The quotient rule helps us find the derivative of a function written as one function divided by another, like our cost function \(C(x) = \frac{u(x)}{v(x)}\). The rule can be written as: \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2} \]
Where:

• \(u(x)\) is the numerator: \(x^3 + 2x + 3\), and \(u'(x) = 3x^2 + 2\) is its derivative.
• \(v(x)\) is the denominator: \(x^2\), and \(v'(x) = 2x\) is its derivative.

By substituting these into the quotient rule formula, we uncover how the cost function behaves as production increases.

Interpreting the Derivative

The interpretation of a derivative is critical in applying calculus to real-world scenarios. In our context, evaluating \(C'(2)\) gives us insight into cost efficiency. With the result \(C'(2) = -0.25\), this tells us that producing one more book, around 2000 copies, reduces the cost per book by $0.25.

• The negative sign indicates a decrease in cost.
• The magnitude shows the rate of cost reduction per additional book.

This type of interpretation helps businesses make strategic production decisions, optimizing their operations to maximize profit while minimizing costs.