Problem 41 - College Algebra and Calculus: An Applied Approach

Question

The number \(y\) of medical degrees conferred in the United States from 1970 through 2004 can be modeled by \(y=0.813 t^{3}-55.70 t^{2}+1185.2 t+7752, \quad 0 \leq t \leq 34\) where \(t\) is the time in years, with \(t=0\) corresponding to 1970. (a) Use a graphing utility to graph the model. Then graphically estimate the years during which the model is increasing and the years during which it is decreasing. (b) Use the test for increasing and decreasing functions to verify the result of part (a).

Step-by-Step Solution

Answer

After graphing the function and analyzing the graph, it can be observed where it is increasing and decreasing. Then by taking the derivative of the function, finding its critical points, and checking the signs of the derivative for each interval, you can mathematically verify the increasing and decreasing periods of the graph. These periods will match the visual representation you derived from observing the graph.

Step 1: a) Graphical Analysis

Firstly, graphing the given function \(y=0.813 t^{3}-55.70 t^{2}+1185.2 t+7752, \; 0 \leq t \leq 34\) using a graphing utility software will give a visual representation of the function. On the graph, observe where the function is going upward (increasing) and where it is going downward (decreasing). Years where the line goes upward represent years when the model is increasing, and where it goes downward show when it is decreasing.

Step 2: b) Test for Increasing and Decreasing Functions

To verify the results derived in step 1, perform a mathematical test. This involves taking the derivative of the function and finding critical points by setting the derivative equal to zero and solving for \(t\). The tested range is \(0 \leq t \leq 34\). Check the signs of the derivative for intervals determined by the critical points. If the derivative is positive, the function is increasing, and if it's negative, the function is decreasing.

Step 3: Take derivative

The derivative of the function \(y\) is: \(\frac{dy}{dt} = 2.439 t^{2} - 111.4 t + 1185.2\).

Step 4: Find critical points

Setting \(\frac{dy}{dt} = 0\) gives the equation: \[2.439 t^{2} - 111.4 t + 1185.2 = 0\]. Solving this quadratic equation will yield two critical points.

Step 5: Check signs of derivative

Now plug in values from each interval determined by the critical points into the derivative. If the result is positive, the function is increasing. If it is negative, the function is decreasing. Do this for all intervals.

Key Concepts

Graphing Utility

Graphing utilities are powerful tools that offer a visual understanding of mathematical models. For example, when graphing the cubic function that models the number of medical degrees conferred over time, graphing software such as Desmos or GeoGebra can provide an immediate and insightful image.

Derivative of a Function

The derivative represents the rate of change of a function. It is a foundational concept in calculus that helps identify how a function behaves. For instance, the derivative of our cubic model, \(2.439 t^{2} - 111.4 t + 1185.2\), informs us about the growth or decline in the number of medical degrees over time.

Critical Points

Critical points are where the derivative of a function equals zero or does not exist These points are crucial for analyzing the function as they indicate potential maximums, minimums, or points of inflection. In our exercise, we solve \(2.439 t^{2} - 111.4 t + 1185.2 = 0\) to find the critical points, which then divide the time interval into sections where we can test for increases or decreases in degrees conferred.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to represent, analyze, and predict real-world phenomena. The exercise presents a model of education trends, with a polynomial function representing the number of medical degrees conferred as a function of time, highlighting the predictive power of calculus in modeling real-world situations.

Calculus in Real-World Applications

Calculus is not just an abstract branch of mathematics but has many practical applications. It helps us understand and predict the dynamics of real-world systems. The principles of calculus, including derivatives and critical points, are used in various fields such as economics, engineering, and environmental science, as seen in the exercise where calculus helps model educational trends over time.