Problem 59 - Brief Applied Calculus

Question

The percentage of people in the United States who are immigrants (that is, were born elsewhere) for different decades is shown below. These percentages are approximated by the function \(f(x)=\frac{1}{2} x^{2}-3.7 x+12\), where \(x\) stands for the number of decades since 1930 (so that, for example, \(x=5\) would stand for 1980 ). a. Find \(f^{\prime}(x)\) using the definition of the derivative b. Evaluate the derivative at \(x=1\) and interpret the result. c. Find the rate of change of the immigrant percentage in the year 2010 .

Step-by-Step Solution

Answer

a. \(f'(x) = x - 3.7\); b. \(-2.7\%\) decrease per decade at \(x=1\); c. \(4.3\%\) increase per decade in 2010.

Step 1: Understand the Derivative Definition

The derivative of a function at a point provides the instantaneous rate of change of the function with respect to the variable. For a given function \(f(x)\), its derivative \(f'(x)\) is defined by the limit \(f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\). We'll apply this to \(f(x) = \frac{1}{2}x^2 - 3.7x + 12\).

Step 2: Compute the Derivative using Definition

Plug \(f(x+h)\) and \(f(x)\) into the derivative definition:\[f'(x) = \lim_{h \to 0} \frac{\left(\frac{1}{2}(x+h)^2 - 3.7(x+h) + 12\right) - \left(\frac{1}{2}x^2 - 3.7x + 12\right)}{h}\]Expand and simplify this expression by canceling out common terms in the numerator.

Step 3: Expand and Simplify the Expression

Expand \((x+h)^2\) to get \(x^2 + 2xh + h^2\). Substituting this into the expression gives:\[f'(x) = \lim_{h \to 0} \frac{\left(\frac{1}{2}(x^2 + 2xh + h^2) - 3.7x - 3.7h + 12\right) - \left(\frac{1}{2}x^2 - 3.7x + 12\right)}{h}\]\[= \lim_{h \to 0} \frac{\frac{1}{2}(2xh + h^2) - 3.7h}{h} \]

Step 4: Simplify Further to Find Derivative

Factor the common factor \(h\) out of the numerator:\[f'(x) = \lim_{h \to 0} \frac{h(x + \frac{1}{2}h) - 3.7h}{h} \]\[= \lim_{h \to 0} \left(x + \frac{1}{2}h - 3.7\right)\]As \(h\) approaches 0, the term \(\frac{1}{2}h\) vanishes, giving:\[f'(x) = x - 3.7\]

Step 5: Evaluate the Derivative at x = 1

Substitute \(x = 1\) into the derivative equation:\[f'(1) = 1 - 3.7 = -2.7\]This indicates that, for \(x=1\), the rate of change of the percentage is decreasing by 2.7% per decade.

Step 6: Find Rate of Change for 2010

For the year 2010, calculate \(x\) by finding the number of decades since 1930. Thus, \(x = \frac{2010 - 1930}{10} = 8\).Substitute \(x = 8\) into the derivative equation:\[f'(8) = 8 - 3.7 = 4.3\]Thus, in 2010, the rate of change of the immigrant percentage is increasing by 4.3% per decade.

Key Concepts

Derivative

The concept of a "derivative" is central to calculus. It helps us understand how a function changes at any given point. If you imagine driving a car, the derivative is like the speedometer telling you how fast the car (or, in math, the function) is going at that exact moment. In mathematical terms, this is known as the "instantaneous rate of change."

For a function like our example, the derivative helps us determine how the percentage of immigrants changes as the years go by. It's found using the formula:

• Take the function, say, (\(f(x) = \frac{1}{2}x^2 - 3.7x + 12\)
• Replace \(x\) by \((x + h)\) in the function.
• Subtract the original function \(f(x)\) from this expression.
• Divide the whole thing by \(h\), and then let \(h\) go to 0. That's the limit part: (\(f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\)

When you work this process out, you end up with a new formula for the rate of change of your function, known as the derivative, \(f'(x) = x - 3.7\). This tells you how quickly the function is changing at each value of \(x\).

Rate of Change

The "rate of change" is simply the speed at which something is changing. In practical terms, think of it as the difference you observe has when moving from one state to another over time. It can be positive, which means an increase, or negative, indicating a decrease.

Using our function for this exercise, the derivative \(f'(x)\) represents the rate of change of the percentage of immigrants over time. For example:

• At \(x=1\) (which stands for 1940), the rate of change is \(-2.7\). This means the percentage of immigrants is decreasing by 2.7% per decade from that point forward.
• For \(x=8\) (the year 2010), the rate of change is \(4.3\). This shows an increase by 4.3% per decade.

This tells us not just how much the percentage is changing, but in which direction and how rapidly that change is occurring at various points in time. Understanding the rate of change is crucial because it reveals trends and patterns within the data that could be critical for future predictions.

Function

A function, in mathematical terms, is essentially a rule that takes an input and gives back an output. It's a lot like a machine that transforms something you put into it (the input) into something else (the output). In our case, the function \(f(x) = \frac{1}{2}x^2 - 3.7x + 12\) uses \(x\) to determine the percentage of immigrants.

Here's how it works:

• "\(x\)" represents the number of decades since 1930. So if \(x = 5\), you're looking at the year 1980.
• The output from the function, \(f(x)\), is the predicted percentage of immigrants for that decade.

The beauty of functions is that they offer a neat and organized way to exhibit relationships between variables. They're very useful for predicting outcomes based on changing conditions. By understanding how the function is defined and how it outputs a result, you can make accurate predictions and adjustments based on real-world data. Functions lie at the heart of both algebra and calculus, making them indispensable tools in understanding how changes happen.