Problem 97 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In a study conducted in 2003 , business spending on technology (in billions of dollars) from the beginning of 2000 through 2005 was projected to be \(S(t)=-1.88 t^{3}+30.33 t^{2}-76.14 t+474 \quad(0 \leq t \leq 5)\) where \(t\) is measured in years, with \(t=0\) corresponding to 2000\. Show that the graph of \(S\) is concave upward on the interval \((0,5)\). What does this result tell you about the rate of business spending on technology over the period in question?

Step-by-Step Solution

Answer

In short, the graph of \(S(t) = -1.88t^3 + 30.33t^2 - 76.14t + 474\) is concave upwards on the interval (0,5) because its second derivative, \(S''(t) = -6(1.88)t + 2(30.33)\), is positive for \(0 \leq t \leq 5\). This indicates that the rate of business spending on technology is increasing from 2000 to 2005, and the rate of growth in business spending accelerates over this period.

Step 1: Calculate the first derivative of S(t)

To calculate the first derivative of S(t), we can apply the power rule to each term: \[ S'(t) = \frac{d}{dt} (-1.88t^3 + 30.33t^2 - 76.14t + 474) \] \[ S'(t) = -3(1.88)t^2 + 2(30.33)t - 76.14 \]

Step 2: Calculate the second derivative of S(t)

Now we will find the second derivative by taking the derivative of the first derivative we found in step 1: \[ S''(t) = \frac{d}{dt} (-3(1.88)t^2 + 2(30.33)t - 76.14) \] \[ S''(t) = -6(1.88)t + 2(30.33) \]

Step 3: Determine the sign of the second derivative S''(t) on the interval (0,5)

To check if the graph of S(t) is concave upward on the interval (0,5), we need to verify whether S''(t) is positive for \(0 \leq t \leq 5\): Let H(t) = S''(t) = \(-6(1.88)t + 2(30.33)\) For t=0, H(0) = \[2(30.33)\] which is positive. For t=5, H(5) = -6(1.88)5 + 2(30.33), which is also positive. Since the second derivative is positive in the given interval (0,5), the graph of S(t) is concave upwards in this interval.

Step 4: Interpret the meaning of concave upward with respect to business spending

The concave upward graph means that the rate of business spending on technology (S'(t)) is increasing throughout the interval from 2000 to 2005. In other words, the rate of growth in business spending on technology accelerates over this period.

Key Concepts

Derivatives

In calculus, derivatives are used to express how a function changes as its input changes. They essentially measure the rate of change or slope of a function. For example, when we look at functions that describe real-world phenomena, the first derivative gives us insight into how one variable affects another.

• The derivative of a function \( f(x) \) is often written as \( f'(x) \) or \( \frac{df}{dx} \).
• The notation implies that derivatives provide a relationship between how outputs vary with inputs.
• For linear functions, derivatives are constant, but for linear combinations or curved graphs, derivatives change according to their mathematical structure.

In the context of the given problem, the first derivative \( S'(t) \) of the spending function \( S(t) \) indicates how business spending on technology changes with time. If this derivative is positive, it indicates that spending is increasing. The calculation of derivatives helps understand the direction and speed at which the expense changes over years.

Concavity

Concavity refers to the shape of the graph of a function. In particular, it focuses on whether the graph is curving upwards or downwards. This can be determined using the second derivative of a function.

• If the second derivative \( f''(x) \) is positive over an interval, the graph is concave upward on that interval.
• Conversely, if \( f''(x) \) is negative, the graph is concave downward.
• A change in the sign of the second derivative may indicate an inflection point, where the graph changes its concavity.

In our exercise, the second derivative \( S''(t) \) determines whether the function \( S(t) \) is concave upward over the interval \((0, 5)\). Since \( S''(t) \) is positive throughout this interval, it tells us that business spending is not only increasing but doing so at an accelerating rate.

Power Rule

The power rule is a fundamental tool in calculus used to find the derivative of functions that are powers of variables. This rule states that if you have a function \( f(x) = x^n \), its derivative is \( f'(x) = nx^{n-1} \). This simplification makes calculating derivatives straightforward, particularly for polynomial expressions.

• The power rule applies to each term of a polynomial individually, making it very efficient for such functions.
• When working with coefficients, remember to multiply the existing coefficient by the exponent before reducing the power of the variable.

In the solution provided, the power rule was used to determine both the first and second derivatives of the spending function \( S(t) \). Taking derivatives step by step with the power rule makes interpreting changes in business spending more approachable, showing us not only the trend but also how sharply the spending increases.