Problem 80 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Almost half of companies let other firms manage some of their Web operations-a practice called Web hosting. Managed services -monitoring a customer's technology services-is the fastest growing part of Web hosting. Managed services sales are expected to grow in accordance with the function $$ f(t)=0.469 t^{2}+0.758 t+0.44 \quad(0 \leq t \leq 6) $$ where \(f(t)\) is measured in billions of dollars and \(t\) is measured in years, with \(t=0\) corresponding to 1999 . a. Find the interval where \(f\) is increasing and the interval where \(f\) is decreasing. b. What does your result tell you about sales in managed services from 1999 through 2005 ?

Step-by-Step Solution

Answer

The function \(f(t)\) is increasing on the interval (0, 6), which indicates that sales of managed services are growing from 1999 through 2005.

Step 1: 1. Find the first derivative of the function

Let us find the first derivative of the given function, \(f(t)\). We will use it to determine the intervals where the function is increasing and decreasing. \[f(t) = 0.469t^2 + 0.758t + 0.44\] Now differentiate with respect to \(t\): \[f'(t) = 2(0.469)t + 0.758\]

Step 2: 2. Find critical points by setting the derivative equal to 0

Now let's set the first derivative \(f'(t)\) equal to 0 and solve for \(t\). These points are critical points of the function, which will help us determine the intervals where the function is increasing and decreasing. \[f'(t) = 2(0.469)t + 0.758 = 0\] Solve for t: \[t = -\frac{0.758}{2(0.469)} \approx -0.808\]

Step 3: 3. Determine the intervals where the function is increasing and decreasing

Since the problem restricts the domain of the function to be between 0 and 6 (0≤ t ≤6), the critical point we found at \(t \approx -0.808\) does not affect the intervals. We now need to determine whether the function is increasing or decreasing in the whole interval \((0, 6)\). We can simply pick any point within the interval (0, 6) and plug it into the first derivative: \[f'(3) = 2(0.469)(3) + 0.758 \approx 3.934 > 0\] The first derivative is positive in the interval (0, 6), which means the function is increasing all over the interval (0, 6). So, \(f\) is increasing on the interval (0, 6).

Step 4: 4. Interpret the result

Based on our result, the sales of managed services (\(f(t)\)) are increasing from 1999 (t=0) to 2005 (t=6), which means that during these years companies are increasingly adopting and spending on managed services.

Key Concepts

First Derivative Test

The First Derivative Test is a handy tool in calculus to determine where a function is increasing or decreasing. By finding the first derivative of a function, we can identify critical points — values of the variable where the derivative is zero or undefined. These points help us determine the behavior of the function.
A positive derivative means the function is increasing. Conversely, a negative derivative indicates the function is decreasing.

• Start by taking the first derivative of your function.
• Find critical points by setting the first derivative equal to zero and solving for the variable.
• Use these critical points to test intervals between them and determine the sign of the derivative.
• Interpret the results to understand where the function increases or decreases.

In our exercise, the function is analyzed between given boundaries, so critical points found outside that range are disregarded.

Increasing Functions

Increasing functions are those where the value of the function grows as the input variable increases. In terms of the derivative, an increasing function has a positive derivative over an interval.
Understanding increasing functions is crucial because it helps predict trends, like sales growth in a business context. To identify whether a function is increasing over an interval, check if the first derivative is greater than zero across that interval.

Steps to determine if a function is increasing:

• Calculate the derivative of the function you are interested in.
• Identify an interval to analyze (in the exercise: the interval is from 0 to 6).
• Plug a test point from that interval into the derivative to check its sign.
• If the derivative is positive, the function is increasing in that interval, as seen with the provided function, which was increasing between 0 and 6.

Increasing functions are indicative of growth, a pattern often desired in contexts such as economic growth or resource consumption.

Quadratic Functions

Quadratic functions are a family of functions represented by the formula \( ax^2 + bx + c \). They create a parabolic graph, with the coefficient \( a \) determining its direction: upward (\( a > 0 \)) or downward (\( a < 0 \)).
In our exercise, the quadratic function exemplifies such a parabola, focusing on positive values that signify an open upward curve.
Essential characteristics of quadratic functions:

• The vertex of the parabola is a crucial point; it determines the maximum or minimum value.
• The axis of symmetry runs vertically through the parabola at the line \( x = -\frac{b}{2a} \).
• The coefficients \(a\), \(b\), and \(c\) affect both the shape and position of the parabola.

Quadratic functions often model natural phenomena and can help in understanding business forecasting, making them a powerful tool in numerous fields.