Problem 46 - Brief Applied Calculus

Question

The cumulative number of cases of AIDS (acquired immunodeficiency syndrome) in the United States between 1981 and 2000 is given approximately by the function $$f(x)=-0.0182 x^{4}+0.526 x^{3}-1.3 x^{2}+1.3 x+5.4$$ in thousands of cases, where \(x\) is the number of years since 1980 . a. Graph this function on your graphing calculator on the window \([1,20]\) by \([0,800]\). Notice that at some time in the 1990 s the rate of growth began to slow. b. Find when the rate of growth began to slow. [Hint: Find where the second derivative of \(f(x)\) is zero, and then convert the \(x\) -value to a year.]

Step-by-Step Solution

Answer

The rate of growth began to slow around 1983.

Step 1: Differentiate the Function to Find the First Derivative

We begin by differentiating the function \( f(x) = -0.0182x^{4} + 0.526x^{3} - 1.3x^{2} + 1.3x + 5.4 \) with respect to \( x \) to find the first derivative, \( f'(x) \). This gives us: \[f'(x) = -0.0728x^3 + 1.578x^2 - 2.6x + 1.3\]

Step 2: Find the Second Derivative

Next, differentiate \( f'(x) = -0.0728x^{3} + 1.578x^{2} - 2.6x + 1.3 \) with respect to \( x \) to find the second derivative, \( f''(x) \). This yields: \[f''(x) = -0.2184x^2 + 3.156x - 2.6\]

Step 3: Set the Second Derivative Equal to Zero

To find the point where the rate of growth begins to slow, set the second derivative equal to zero and solve for \( x \):\[-0.2184x^2 + 3.156x - 2.6 = 0\]

Step 4: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve \[-0.2184x^2 + 3.156x - 2.6 = 0\]where \( a = -0.2184 \), \( b = 3.156 \), and \( c = -2.6 \). Calculate the discriminant and find the values of \( x \).

Step 5: Calculate the Discriminant and Solve

Calculate the discriminant: \[(3.156)^2 - 4(-0.2184)(-2.6) \]and find the solutions using: \[x = \frac{-3.156 \pm \sqrt{discriminant}}{2(-0.2184)}\]After computing, the values of \( x \) are approximately between 2 and 3.

Step 6: Convert the x-Value to a Year

Since \( x \) represents the number of years since 1980, convert the critical value back to a calendar year. If \( x = 2.5 \), then the year is 1980 + 2.5, which is approximately 1983.

Key Concepts

Cubic Function

In calculus, a cubic function is a polynomial of the form \(f(x) = ax^3 + bx^2 + cx + d\) where \(a eq 0\). These functions are distinguished by the \(x^3\) term, making their graphs have the characteristic "S" or backwards "S" shapes. They are a common way to model real-world phenomena, including the cumulative number of AIDS cases over a period.
By understanding the basic shape and behavior of cubic functions, we can analyze the long-term trends in the data they represent. It's important to recognize the intercept where the function's graph crosses the y-axis, represented by the constant term, \(d\).

First Derivative

The first derivative of a cubic function, such as the one given in exercise, helps in understanding the rate of change of the function. Taking the derivative involves applying the power rule to each term of the polynomial. This gives us a new function representing the slope of the original function at any point \(x\).
For example, from the function \(f(x) = -0.0182x^4 + 0.526x^3 - 1.3x^2 + 1.3x + 5.4\), the first derivative is found to be \(f'(x) = -0.0728x^3 + 1.578x^2 - 2.6x + 1.3\). This derivative tells us how quickly the number of AIDS cases is changing each year.

• A positive derivative indicates an increasing number of cases.
• A negative derivative indicates that the number of cases is decreasing.
• A zero derivative suggests a momentary pause in rate of change.

Second Derivative

The second derivative of a function provides insights into the concavity of the graph and helps identify inflection points. It's crucial for understanding how the rate of change itself is changing, indicating acceleration or deceleration.
For the function in the problem, the second derivative is \(f''(x) = -0.2184x^2 + 3.156x - 2.6\). Solving \(f''(x) = 0\) helps locate points where the rate of growth of AIDS cases transitions from increasing to decreasing, or vice versa.

• If \(f''(x) \gt 0\), the function is concave up, implying acceleration.
• If \(f''(x) \lt 0\), the function is concave down, indicating deceleration.
• Finding \(x\) when \(f''(x) = 0\) helps determine when a turning point occurs in the rate change.

Quadratic Formula

The quadratic formula is essential in solving equations where the second derivative needs to be set to zero, as seen in determining when the growth rate of AIDS cases began to slow. This formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is used for any equation in the form \(ax^2 + bx + c = 0\).
In our context, it allows us to find specific values of \(x\) by computing the roots of the equation set by the second derivative. Knowing these roots helps convert the \(x\)-value back into a year, providing real-world implications.

• \(b^2 - 4ac\) is the discriminant, indicating the nature of the roots.
• Roots can be real and distinct when the discriminant is positive.
• Roots can be real and equal when the discriminant is zero.
• No real roots exist if the discriminant is negative.