91Ó°ÊÓ

In an economic enterprise, the total amount \(T\) that is produced is a function of the amount \(n\) of a given input used in the process of production. For example, the yield of a crop depends on the amount of fertilizer used, and the number of widgets manufactured depends on the number of workers. Because of the law of diminishing returns, a graph for \(T\) commonly has an inflection point followed by a maximum, so a cubic model may be appropriate. In this exercise we use the model $$ T=-2 n^{3}+3 n^{2}+n $$ with \(n\) measured in thousands of units of input and \(T\) measured in thousands of units of product. a. Make a graph of \(T\) as a function of \(n\). Include values of \(n\) up to \(1.5\) thousand units. b. Express using functional notation the amount produced if the input is \(1.45\) thousand units, and then calculate that value. c. Find the approximate location of the inflection point and explain what it means in practical terms. d. What is the maximum amount produced?

Step by step solution

01

Graph the Function

To graph the function, we plot \( T = -2n^3 + 3n^2 + n \) for \( n \) from 0 to 1.5. We calculate values of \( T \) at various points like \( n = 0, 0.5, 1.0, 1.5 \), and sketch the curve using these points to illustrate the behavior of the cubic function.

02

Calculate for a Specific Input

For part b, we need \( T(1.45) \). Substitute \( n = 1.45 \): \( T(1.45) = -2(1.45)^3 + 3(1.45)^2 + 1.45 \), which gives \( T(1.45) \). Evaluating this expression, we get the amount produced when the input is 1.45 thousand units.

03

Locate the Inflection Point

The inflection point occurs where the second derivative changes sign. Start by finding the first derivative \( T'(n) = -6n^2 + 6n + 1 \) and then the second derivative \( T''(n) = -12n + 6 \). Set \( T''(n) = 0 \) to find \( n \), giving \( n = 0.5 \) thousand units. This is our inflection point.

04

Interpret the Inflection Point

The inflection point \( n = 0.5 \) implies a location where the rate of increase of production starts to decrease. Before \( n = 0.5 \), production increases at an increasing rate; after this, production increases at a decreasing rate.

05

Find the Maximum Production

To find the maximum, set the first derivative \( T'(n) = -6n^2 + 6n + 1 \) to zero and solve for \( n \). This gives the critical points. Solving, we find \( n \approx 1.195 \). Evaluating \( T(1.195) \), we find the maximum production value.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inflection Point

An **inflection point** is where the curve of a function changes its nature from concave (curving upward) to convex (curving downward), or vice versa. In terms of derivatives, the inflection point is where the second derivative of the function changes sign.
In the cubic function model for the production given by \[ T = -2n^3 + 3n^2 + n \]we locate this point by finding the second derivative \(T''(n)\) and setting it to zero:\[ T''(n) = -12n + 6 \]Solving \( -12n + 6 = 0 \) results in \( n = 0.5 \) thousand units. At this point, the curve switches from increasing its growth rate to decreasing its growth rate.
This change indicates a crucial strategic point in production processes. Prior to the inflection point, adding more input increases the output rate more significantly. Beyond it, added inputs continue to increase output, but at a slowing rate, signaling a shift towards diminishing productivity.

Law of Diminishing Returns

The **law of diminishing returns** is a key economic principle that explains how successive increases in inputs lead to smaller incremental gains in outputs. It plays an instrumental role in understanding production processes, especially when using a cubic function.
In the context of the given function, \[ T = -2n^3 + 3n^2 + n \]the law of diminishing returns suggests that after a certain point, each additional unit of input yields less and less product. This is visually represented in the graph by the flattening of the curve after the inflection point, eventually peaking at a maximum point.

• Before the inflection point, output grows at an accelerating rate.
• Beyond the inflection point, returns diminish, meaning each additional input contributes less to the total output.

Understanding this law helps businesses and economists determine optimal production levels and adjust resources to avoid inefficient overuse.

Functional Notation

**Functional notation** is a way of representing the relationship between variables in a mathematical model. It allows for a concise way to express complex equations and is crucial in analyzing how a function behaves.
In our exercise, the production model is denoted as\[ T(n) = -2n^3 + 3n^2 + n \]where \( T(n) \) represents the output (in thousands of units) based upon the input \( n \) (also in thousands of units). Functional notation simplifies the expression and calculation of outputs for specific inputs.
For example, to find the production level at an input level of 1.45 thousand units, we use:\[ T(1.45) = -2(1.45)^3 + 3(1.45)^2 + 1.45 \]Calculating this gives the exact production at that specific input value. Functional notation is invaluable as it provides a straightforward means to compute specific outputs, track relationships, and analyze changes within the model.

First and Second Derivatives

Derivatives offer insights into the behavior of a function, especially in predicting how small changes in input affect the output.

• The **first derivative**, \( T'(n) \), tells us the rate at which the total output changes with respect to changes in the input. For the given function, \[ T'(n) = -6n^2 + 6n + 1 \] it captures how the production increases or decreases as more input is used.
• The **second derivative**, \( T''(n) \), indicates the acceleration or deceleration of that rate. It can point out whether the function is speeding up or slowing down in its increase or decrease. For example, \[ T''(n) = -12n + 6 \] changing sign at \( n = 0.5 \) tells us that this point is an inflection, highlighting a transition in the rate of change.

These derivatives help not only in calculating critical points such as maxima and inflection points but also in understanding the dynamics of the system modeled by the function. Having a grasp of derivatives assists significantly in optimization and efficiency-driven decisions.