Problem 6 - Calculus for Biology and Medicine

Question

Let $$ f(x)=\sqrt{x}, \quad x \geq 0 . $$ (a) Graph \(y=f(x)\) for \(0 \leq x \leq 6\). (b) For which values of \(x\) is \(y=f(x)\) within \(0.2\) of \(2 ?\) (Hint: Find values of \(x\) such that \(|\sqrt{x}-2|<0.2 .)\) (c) Illustrate your result in (b) on the graph that you obtained in (a).

Step-by-Step Solution

Answer

The values of \( x \) where \( y = \sqrt{x} \) is within 0.2 of 2 are \( 3.24 < x < 4.84 \).

Step 1: Graphing the function

To graph the function \( y = \sqrt{x} \), we need to calculate the values of \( y \) for \( 0 \leq x \leq 6 \). Calculate a few points: \( (0, 0), (1, 1), (4, 2), (6, \sqrt{6}) \), and then sketch the curve. The curve will start at the origin and increase gradually, making a smooth, gentle upward curve.

Step 2: Set up inequality for solution

We want to find all \( x \) such that \( |\sqrt{x} - 2| < 0.2 \). This can be rewritten as two separate inequalities: \( \sqrt{x} - 2 < 0.2 \) and \( \sqrt{x} - 2 > -0.2 \).

Step 3: Solve first inequality

Start with \( \sqrt{x} - 2 < 0.2 \). This implies \( \sqrt{x} < 2.2 \). Squaring both sides, we have \( x < 4.84 \).

Step 4: Solve second inequality

Now consider \( \sqrt{x} - 2 > -0.2 \). This implies \( \sqrt{x} > 1.8 \). Squaring both sides, we have \( x > 3.24 \).

Step 5: Combine solutions

The solution to \( |\sqrt{x} - 2| < 0.2 \) is the intersection of \( x < 4.84 \) and \( x > 3.24 \). Thus, \( 3.24 < x < 4.84 \).

Step 6: Illustrate results on graph

On the graph created in Step 1, highlight the section of the curve where \( 3.24 < x < 4.84 \). This is the part of the graph where the function \( y = \sqrt{x} \) is within \( 0.2 \) of \( y = 2 \), and should be shaded or marked distinctly.

Key Concepts

Graphing functions

When we graph functions, we're essentially creating a visual representation of how a function behaves. It lets us see how the values change and relate to each other as we move along the x-axis. For a function like \( y = \sqrt{x} \), the process starts by plotting points. This is because the graph is a collection of all the possible pairs of \( x \) and their corresponding \( y \) values.

• Begin at the point \((0,0)\) because the square root of zero is zero.
• As \( x \) increases, calculate additional points, such as \((1,1)\) since \( \sqrt{1} = 1 \).
• Continue with points like \((4,2)\) and \((6, \sqrt{6})\) to see the trend.

The curve of this function shows an upward trend, starting from the origin and moving smoothly to the right. Unlike linear functions, which have straight lines, this curve gradually flattens and opens wide as \( x \) increases. This characteristic is typical of square root functions, reflecting their nature to increase slowly after initially growing rapidly.

Inequalities

Inequalities are a way to express that one quantity is less than or greater than another. In the context of our problem, we want to determine when the function \( y = \sqrt{x} \) stays within a certain distance from a specified value, like \( y = 2 \). The equation \( |\sqrt{x} - 2| < 0.2 \) means we're interested in all \( x \) values where the output of the function is between 1.8 and 2.2. To do this, we analyze two separate inequalities:

• \( \sqrt{x} - 2 < 0.2 \), which means \( \sqrt{x} < 2.2 \).
• \( \sqrt{x} - 2 > -0.2 \), which results in \( \sqrt{x} > 1.8 \).

Once solved, these inequalities define the specific range \( 3.24 < x < 4.84 \), where the function stays close to 2. By visualizing this range on the graph, we can clearly see where the square root function dips below and rises above the line \( y = 2 \), giving us a practical sense of the function's behavior between these bounds.

Square root functions

Square root functions, such as \( f(x) = \sqrt{x} \), have a unique shape. These functions only take non-negative values for \( x \) because square roots of negative numbers aren't real numbers. The function starts at the origin and grows at a decreasing rate, meaning it rises quickly at first before gradually leveling out as \( x \) increases.
Some properties of square root functions include:

• They only operate in the first quadrant of a graph, where both \( x \) and \( y \) values are non-negative.
• They are continuous and smooth, without any breaks or sharp corners.
• Asymptotically, they approach infinity, meaning they keep increasing but at a slower pace.

By understanding these traits, one can predict the function's long-term behavior, noticing how it stretches out over the x-axis while steadily climbing upwards. This makes them crucial in various applications like physics and engineering, where understanding gradual growth and constraint-based solutions is key.