Problem 100 - Algebra and Trigonometry

Question

Find the average rate of change of \(f(x)=\sqrt{x}\) from \(x_{1}=4\) to \(x_{2}=9 .\) (Section \(2.4,\) Example 4 )

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x) = \sqrt{x}\) over the interval from \(x_{1} = 4\) to \(x_{2} = 9\) is 0.2

Step 1: State the Formula

The formula for the average rate of change is \(AvgRate = \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}\)

Step 2: Plug in the Values

Next, substitute \(x_{1}=4\) and \(x_{2}=9\) into the function \(f(x)=\sqrt{x}\) over the interval. This gives us \(f(x_{1}) = \sqrt{4} = 2\) and \(f(x_{2}) = \sqrt{9} = 3\)

Step 3: Calculate the Average Rate of Change

After substituting the values of \(f(x_{1})\) and \(f(x_{2})\) into the rate of change formula, we get \(AvgRate = \frac{3 - 2}{9 - 4} = \frac{1}{5} = 0.2\)

Key Concepts

Functions

When we talk about functions in math, we're essentially talking about a relationship between two sets of numbers. A function takes an input (usually called \( x \)) and produces an output (called \( f(x) \)). You can think of it like a machine where you feed in a number and get another number out.
For example, in our exercise, the function is \( f(x) = \sqrt{x} \). This means if you put a number into the function (that is, substitute for \( x \)), it gives you the square root of that number. Functions can be represented in various forms such as equations, graphs, or tables.

• Equations show a relationship in a concise mathematical way.
• Graphs visually show how the function behaves over a range of values.
• Tables can list inputs and their corresponding outputs.

Understanding functions helps us describe and model real-world relationships and changes.

Square Roots

A square root is a special mathematical function. It is the opposite of squaring a number. If you square the number 3, you multiply it by itself to get 9. The square root of 9 is 3. It's useful to remember that when you see \( \sqrt{x} \), you are looking for a number which, when squared, will give \( x \).
In the example, we have \( \sqrt{4} = 2 \) because 2 squared equals 4, and \( \sqrt{9} = 3 \) because 3 squared equals 9.
It's also important to note:

• The square root of a positive number is always positive in basic arithmetic.
• Square roots of non-perfect squares result in irrational numbers (numbers that can't be perfectly written as fractions).

Square roots are fundamental in various areas of math and physics, making them critical to understand.

Interval Notation

Interval notation is a way of writing the set of numbers that lie between two endpoints. It helps succinctly describe a range, such as the portion of a number line that a function is interested in. There are some symbols to be aware of:

• Square brackets \([ ]\) are used when the endpoint is included (inclusive).
• Parentheses \(( )\) are used when the endpoint is not included (exclusive).

For instance, the interval \([4, 9]\) includes both 4 and 9. In this context, the interval is the range over which we calculate the average rate of change for our function. Understanding interval notation is crucial because it allows mathematicians to clearly specify parts of functions they are describing or analyzing.

Algebraic Expressions

Algebraic expressions are a combination of variables, numbers, and operators (like plus or minus signs). They express a particular quantity mathematically.

• An example is \( \sqrt{x} \), where \( x \) is the variable.
• Expressions can be combined to describe more complex relationships.

Algebraic expressions often represent real-world situations or unknown values to find solutions. In our exercise, we use the expression \( \sqrt{x} \) to describe how the function behaves as \( x \) changes. Knowing how to work with algebraic expressions helps us solve equations and analyze mathematical relationships effectively.