Problem 101 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The amount (in billions of dollars) spent by the top 15 U.S. financial institutions on IT (information technology) offshore outsourcing is projected to be $$ A(t)=0.92(t+1)^{0.61} \quad(0 \leq t \leq 4) $$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 2004 . a. Show that \(A\) is increasing on \((0,4)\) and interpret your result. b. Show that \(A\) is concave downward on \((0,4)\). Interpret your result.

Step-by-Step Solution

Answer

The function \(A(t) = 0.92(t + 1)^{0.61}\) is increasing on the interval (0, 4), which means that the amount spent on IT offshore outsourcing is projected to grow over the next four years. However, since \(A(t)\) is concave downward on the interval (0,4), the rate of growth would decrease over time.

Step 1: Find the first derivative of A(t).

We will find the derivative of the function \(A(t) = 0.92(t + 1)^{0.61}\) with respect to t: \(A'(t) = 0.92 \cdot 0.61 (t+1)^{-0.39} \)

Step 2: Determine if A'(t) is positive on (0,4).

Now, we need to check if \( A'(t) > 0 \) on the interval (0, 4). \( A'(t) = 0.92 \cdot 0.61 (t+1)^{-0.39} \) Since \(0.92\) and \(0.61\) are both positive constants, we just need to determine the sign of \((t+1)^{-0.39}\) on the interval (0,4). Since the exponent is negative, it means that the quantity will always be positive regardless of the value of \(t\). Thus, the product of these constants and the positive quantity will also be positive. Therefore, \(A'(t) > 0\) on the interval (0, 4). Interpretation: Since \(A'(t) > 0\), the function A(t) is increasing on the interval (0,4). Meaning that the amount spent on IT offshore outsourcing is projected to grow over the next four years.

Step 3: Find the second derivative of A(t).

To analyze the concavity of A(t), we'll find the second derivative of A(t) with respect to t: \(A''(t) = 0.92 \cdot 0.61 \cdot (-0.39) (t+1)^{-1.39}\)

Step 4: Determine if A''(t) is negative on (0,4).

Now, we check whether \(A''(t) < 0\) on the interval (0, 4). \(A''(t) = 0.92 \cdot 0.61 \cdot (-0.39) (t+1)^{-1.39}\) Since 0.92 and 0.61 are positive constants and the exponent -1.39 is negative, and the term (-0.39) is negative, the quantity \((t+1)^{-1.39}\) will always be positive. The product of these constants and the term (-0.39) will be negative, so \(A''(t) < 0\) on the interval (0,4). Interpretation: Market is showing that the amount spent on IT offshore outsourcing is projected to grow over the next four years but the rate of growth would decrease over time as A(t) is concave downward on the interval (0,4).

Key Concepts

First Derivative Test

The first derivative test is a useful tool in calculus to determine whether a function is increasing or decreasing on certain intervals. By finding the first derivative of a function and analyzing its sign, we can infer the direction of the function's slope over specific ranges.

In our exercise, the first derivative of the offshore outsourcing spending function, denoted as \(A'(t)\), was found and determined to be positive over the interval (0, 4). This indicates that within those four years, the spendings are continuously rising. Since the derivative \(A'(t)\) is greater than zero, we conclude that the function \(A(t)\) is increasing—signifying an upward trend in IT outsourcing spendings by these financial institutions.

Second Derivative Test

The second derivative test comes into play when we want to understand the concavity of a function or, put simply, how the function curves. It involves taking the second derivative of our original function and determining its sign. For a function with a negative second derivative, the concavity is downward, meaning it bends like a frown.

By computing the second derivative \(A''(t)\) for the spending function and noting that it is negative across the interval (0, 4), we can infer that while the spending is indeed increasing, it's doing so at a decreasing rate. This could suggest that initial investments in IT offshore outsourcing are significant and subsequently, over the years, either the investments are maturing or becoming more efficient, requiring less financial input to maintain or grow.

Exponential Growth

Exponential growth refers to an increase that is proportional to the current amount. In other words, the rate of growth is consistent relative to the size of the function's value. The function we have, however, does not characterize pure exponential growth as it would if the function was expressed as \(A(t) = Pe^{rt}\), where \(P\) is the initial amount, \(r\) is the growth rate, and \(t\) is time.

Though our function \(A(t) = 0.92(t+1)^{0.61}\) does show growth, it doesn't grow exponentially because the exponent here is less than 1. This means the amount spent is increasing but it does not double at a consistent interval, which is typical for exponential growth. This distinction is crucial for students to recognize the difference between various types of growth patterns in mathematical modeling.

Mathematical Modeling

Mathematical modeling is a process of using mathematical structures and concepts to represent real-world scenarios. It allows us to predict and analyze behaviors and patterns using functions, equations, and other analytical tools.

In this scenario, the function \(A(t) = 0.92(t+1)^{0.61}\) serves as a model for predicting the spending on IT offshore outsourcing by U.S. financial institutions over a four-year period. It encapsulates the projected growth trend and provides a simplified yet powerful representation of how spending is expected to change over time. Understanding how to construct and interpret these models is a critical skill for students, especially when venturing into fields that involve data analysis and forecasting.