Problem 16 - Calculus Concepts

Question

School Enrollment The function \(E\) gives the public secondary school enrollment, in millions of students, in the United States between 1940 and \(2008 .\) The input \(x\) represents the number of years since \(1940 .\) Use the following information to sketch a graph of \(E\). (Sources: Based on data appearing in Datapedia of the United States, Lanham, MD: Bernan Press, 1994 ; and in Statistical Abstract, 1998 and 2009\()\) \- \(E(40)=13.2\) \- The graph of \(E\) is always concave down. \- Between 1980 and 1990 , enrollment declined at an average rate of 0.19 million students per year. \- The projected enrollment for 2008 is 14,400,000 students. \- It is not possible to draw a line tangent to the graph of \(E\) at \(x=50\)

Step-by-Step Solution

Answer

Plot points at \((40, 13.2)\), \((50, 11.3)\), \((68, 14.4)\); ensure graph is concave down, with a cusp at \(x = 50\).

Step 1: Identify Key Points

The function \(E\) gives the enrollment in millions. Start with the point from 40 years since 1940. This gives us the point \((40, 13.2)\). This means in 1980 (since \(1940 + 40 = 1980\)), the enrollment was 13.2 million students. At the point \(x = 68\), which translates to the year 2008, the projected enrollment is 14.4 million students, giving us the point \((68, 14.4)\).

Step 2: Consider Concavity

The graph is always concave down. This means that as we move from left to right, the slope of the tangent to the function \(E\) is decreasing. The graph opens downward like an upside-down parabola or similar curve.

Step 3: Understand Rate of Change Between 1980 and 1990

Between 1980 (\(x=40\)) and 1990 (\(x=50\)), the enrollment decreases at a rate of 0.19 million students per year. The decrease over 10 years is \(0.19 \times 10 = 1.9\) million. Thus, \(E(50) = 13.2 - 1.9 = 11.3\), giving the point \((50, 11.3)\).

Step 4: Consider the Tangency at x = 50

It is not possible to draw a tangent line at \(x=50\), suggesting a potential cusp or vertical tangent behavior. This implies a transition point in the graph’s curvature or slope at this position.

Step 5: Sketch the Graph

Plot the points \((40, 13.2)\), \((50, 11.3)\), and \((68, 14.4)\) on a coordinate grid. Ensure the graph is concave down throughout. Between \(x = 40\) and \(x = 50\), there is a noticeable downward trend in enrollment. After \(x = 50\), the trend moves upwards but still respects the concave down condition. Include a potential cusp at \(x = 50\) as the tangency is undefined here.

Key Concepts

Concavity

Concavity is a concept in calculus that describes how a curve bends. When we say that a graph is "concave down," it means that as you move from left to right along the curve, it bends downwards like a frown. Concave down curves have the property that any tangent line to the curve will lie above the graph. This is crucial for the function given in the exercise, as it tells us the overall shape of the graph:

• The slope of the tangent line decreases as you move from left to right.
• The graph resembles an upside-down parabola or similar curves that open downward.

Understanding concavity helps us visualize the curve even without detailed calculations.
For the school enrollment function, knowing it is always concave down provides insight into overall trends such as decreasing rates of change even as values rise.

Rate of Change

The rate of change is a fundamental concept in calculus, particularly useful in understanding how quickly something is changing over a particular interval. It can be thought of as the "speed" at which the dependent variable (e.g., school enrollment) is changing with respect to the independent variable (e.g., time in years). In this context, it is given as an average rate over a specified period:

• Between 1980 and 1990, the enrollment decreased by 0.19 million students per year on average.
• This means that if you plot the enrollment numbers, each year during that decade, the graph would show a gradual decrease.

The rate of change provides detailed insight into specific trends within the wider time span of the data. It helps in sketching the graph accurately by pointing out key intervals where the rate of change differs from the rest.

Tangent Line

A tangent line is a straight line that touches a curve at only one point and represents the instantaneous rate of change of the function at that point. In simpler terms, it's like placing a ruler on the curve without cutting through it. The slope of this line tells us how steep the curve is at that specific point:

• In the given exercise, it is stated that no tangent line can be drawn at \(x=50\).
• This suggests unusual behavior at that point, such as a cusp or vertical tangent, where the standard definition of slope (change in y over change in x) potentially becomes undefined.

Understanding the behavior of tangent lines is critical to analyzing the curve’s geometry and behavior. It helps students recognize where the behavior of the graph changes dramatically, guiding expectations about the curve's local behavior.