Problem 48 - Single Variable Calculus

Question

Spending on Fiber-0ptic Links U.S. telephone company spending on fiber-optic links to homes and businesses from the beginning of 2001 to the beginning of 2006 is approximated by \(S(t)=-2.315 t^{3}+34.325 t^{2}+1.32 t+23 \quad 0 \leq t \leq 5\) billion dollars in year \(t\), where \(t\) is measured in years with \(t=0\) corresponding to the beginning of 2001 . a. Plot the graph of \(S\) in the viewing window \([0,5] \times[0,600]\) b. Plot the graph of \(S^{\prime}\) in the viewing window \([0,5] \times[0,175]\). What conclusion can you draw from your result? c. Verify your result analytically.

Step-by-Step Solution

Answer

The short answer based on the provided solution is: a. Plot the graph of \(S(t) = -2.315 t^{3}+34.325 t^{2}+1.32 t+23\) on the viewing window \([0,5] \times [0,600]\) to visualize spending on fiber-optic links. b. Find the derivative \(S'(t) = -6.945 t^{2} + 68.65t + 1.32\) and plot it on the viewing window \([0,5] \times [0,175]\). Draw conclusions based on the sign of \(S'(t)\) to understand how spending on fiber-optic links is changing. c. Verify the result analytically by finding the critical and inflection points of \(S(t)\). Solve the equations \(-6.945 t^{2} + 68.65t + 1.32 = 0\) for critical points and \(-13.89t + 68.65 = 0\) for inflection points. Analyze the behavior of the rate of spending on fiber-optic links over time using this information.

Step 1: a. Plot the function \(S(t)\)

To plot the graph of \(S(t)\), we first recall that the function for spending on fiber-optic links is given by: \(S(t)=-2.315 t^{3}+34.325 t^{2}+1.32 t+23\) To plot the graph, use graphing software or calculator with the viewing window of \([0,5] \times [0,600]\). This will produce the graph showing the spending on fiber-optic links as a function of time.

Step 2: b. Find derivative and plot \(S'(t)\)

To analyze how the spending on fiber-optic links is changing over time, we will find the derivative \(S^{\prime}(t)\) with respect to time \(t\). Differentiating \(S(t)\), we get: \(S^{\prime}(t) = -6.945 t^{2} + 68.65t + 1.32\) Now, plot the graph of \(S^{\prime}(t)\) using the same graphing software or calculator, using the viewing window of \([0,5] \times [0,175]\). The graph will give us an understanding of how the rate of spending on fiber-optic links is changing over the interval \(0 \leq t \leq 5\). Based on the graph, if \(S^{\prime}(t)\) is positive, that would mean the spending is increasing. On the other hand, if it is negative, that would mean the spending is decreasing over time. Assess the sign of the graphed function over the given interval to draw necessary conclusions on the change of spending trend over time.

Step 3: c. Analyzing critical and inflection points

To verify the previous result analytically, we will first find the critical points (i.e., where the derivative is zero or undefined) and analyze whether there are any relative maxima or minima in the function \(S(t)\). Additionally, we will find the second derivative of \(S(t)\) to characterize the inflection points. Finding the critical points: Set \(S^{\prime}(t) = 0\): \(-6.945 t^{2} + 68.65t + 1.32 = 0\) This is a quadratic equation that can be solved using quadratic formula, factoring or numerical techniques to find the values of \(t\) for which the derivative is zero. Next, find the second derivative, \(S^{\prime\prime}(t)\): \(S^{\prime\prime}(t) = -13.89t + 68.65\) Now, set \(S^{\prime\prime}(t) = 0\): \(-13.89t + 68.65 = 0\) Solve for \(t\) to find the potential inflection points. By conducting this analysis, you can verify the behavior of the rate of spending on fiber-optic links over time analytically.

Key Concepts

Derivatives

Derivatives show us how a function changes as its input changes. In this context, the derivative of the spending function, denoted as \(S'(t)\), helps us understand the rate at which spending is increasing or decreasing over time. To find it, we differentiate the original function \(S(t)\):

• The derivative \(S'(t) = -6.945t^2 + 68.65t + 1.32\) tells us how fast the spending is changing at any given year \(t\).
• If \(S'(t)\) is positive, spending is increasing. If it's negative, spending is decreasing.

This tool is vital in both mathematics and real-life scenarios, as it translates a static view of a function into dynamic insights about change.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to whole number powers and their coefficients. The function \(S(t) = -2.315t^3 + 34.325t^2 + 1.32t + 23\) is a polynomial. Here's what you should know about them:

• This function is a cubic polynomial due to the highest power of \(t\), which is 3.
• Polynomials can model a wide range of real-life situations, like spending over time, as seen here.
• They are smooth and continuous, making them ideal for modeling trends and predictions.

Understanding these functions is key when analyzing complex relationships in fields like economics and engineering.

Graphing

Graphing a function helps visualize how it behaves across different values. For \(S(t)\), graphing shows spending from 2001 to 2006.

• The graph of \(S(t)\) should be plotted in the window \([0,5] \times [0,600]\), displaying the polynomial's progression over years.
• Turning to \(S'(t)\), plot it in \([0,5] \times [0,175]\). This graph reveals how the spending rate changes over time.

Interpreting these graphs means understanding where spending rises, falls, or stabilizes. It's a visual way to see abstract algebraic information in an understandable form.

Critical Points

Critical points occur where the derivative equals zero. They tell us where a function might change direction, indicating potential maxima or minima.

• For \(S'(t)\), setting it to zero helps locate the critical points: \(-6.945t^2 + 68.65t + 1.32 = 0\).
• Solve this quadratic equation to find the exact years where these critical changes occur.

Moreover, by calculating the second derivative \(S''(t) = -13.89t + 68.65\), we gain insight into the concavity of \(S(t)\). Setting \(S''(t) = 0\) gives us inflection points that highlight where the graph's curvature changes.
This combined analysis offers a clearer picture of spending trends over time.