Problem 58 - Brief Applied Calculus

Question

Graph the cost function \(y_{1}=\sqrt{4 x^{2}+900}\) on the window \([0,30]\) by \([-10,70]\). Then use NDERIV to define \(y_{2}\) as the derivative of \(y_{1}\). Verify the answer to Exercise 57 by evaluating the marginal cost function \(y_{2}\) at \(x=20\).

Step-by-Step Solution

Answer

Graph \(y_{1} = \sqrt{4x^{2} + 900}\), define \(y_{2}\) using NDERIV. Evaluate \(y_{2}(20)\) for marginal cost.

Step 1: Understanding the Graphing Window

The graphing window described is a region on the coordinate plane that will display the curve. The horizontal window is set from 0 to 30, and the vertical window is set from -10 to 70. This means we will plot the cost function within these bounds for a better view of its behavior.

Step 2: Plotting the Cost Function

The cost function given is \(y_{1} = \sqrt{4x^{2} + 900}\). To plot this function, you will graph it within the window specified. The function is a square root function, and it primarily shows how cost changes with respect to the independent variable \(x\). Use a graphing calculator or graphing software to plot the function within the given window to identify its shape and trends.

Step 3: Using NDERIV to Find the Derivative Function

The NDERIV feature on graphing calculators like the TI-series computes the derivative of a function symbolically. Define \(y_{2}\) as the derivative of \(y_{1}\) by inputting the expression for \(y_{1}\) into the calculator and using the NDERIV function. This will give the expression for the marginal cost or the rate of change of the cost with respect to \(x\).

Step 4: Evaluating the Marginal Cost at \(x=20\)

Substitute \(x=20\) into the derivative function \(y_{2}\). Use either the expression obtained or the calculator to find the value of the marginal cost at this point. This calculation will confirm the sensitivity or instantaneous rate of increase of cost when \(x=20\).

Step 5: Comparing with Exercise 57

After obtaining the value of \(y_{2}\) at \(x=20\), compare it with the result obtained in Exercise 57 to ensure consistency and accuracy. This will also validate the correctness of your computations from previous parts of the problem.

Key Concepts

Cost Function

Cost functions are essential in understanding how costs change with different levels of production or variables represented by the independent variable, typically denoted as \(x\). In this case, the cost function is \(y_1 = \sqrt{4x^2 + 900}\). This specific function reflects a relationship where cost changes as the square root of an expression involving \(x^2\).
Such functions are used widely in economics and business to represent production costs, including fixed and variable costs. The square root nature suggests that cost grows but at a decreasing rate as \(x\) increases. When graphing, the cost function gives insights into how expenses will evolve as production or some dependent variable changes.

Derivative Calculation

The derivative of a function is a fundamental concept in calculus. It provides the rate at which a function changes concerning its independent variable. For the cost function \(y_1 = \sqrt{4x^2 + 900}\), the derivative, denoted as \(y_2\), is crucial in understanding how cost changes at a specific point.
Calculating derivatives analytically involves using rules of differentiation, such as the chain rule, which is applicable here due to the combination of a square root. However, a graphing calculator offers a feature called NDERIV, simplifying this process. When you define \(y_2\) as the derivative of \(y_1\) with the NDERIV function, it outputs a new graph which represents how quickly costs are changing at each point on the \(x\)-axis.

Marginal Cost

Marginal cost is a concept that derives from the derivative calculation. It represents the additional cost incurred by producing one more unit, or more generally, the cost of a small increase in the quantity of \(x\). For the function \(y_1 = \sqrt{4x^2 + 900}\), its derivative function \(y_2\) acts as the marginal cost.
When evaluating this marginal cost at \(x = 20\), you substitute 20 into the derivative expression to calculate how cost is changing at that specific point. This insight is critical for decision-making, as it tells businesses how their costs will respond to changes in production. For this exercise, checking the marginal cost at \(x = 20\) allows for validation against previous findings (like those in Exercise 57) to ensure accuracy.

Graphing Calculator Techniques

Utilizing a graphing calculator effectively is a vital skill in solving complex calculus problems. These calculators have functions like NDERIV, which streamline finding derivatives—a process vital for cost and marginal cost analysis.
In this exercise, after plotting \(y_1 = \sqrt{4x^2 + 900}\) within the defined window, employing the NDERIV feature computes the derivative. To master this, familiarizing oneself with menu navigation and input syntax of the calculator is important.
Additionally, graphing calculators allow users to visually analyze function behavior and derivatives, providing insights beyond mere algebraic solutions. They facilitate evaluating values like \(y_2\) at \(x=20\), showing how numeric computations corroborate with theoretical exercises learned in class.