Problem 2 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(1-6,\) find the average rate of change of the function over each interval. \(f(x)=\sqrt{4 x+1}\) (a) $$[0,2] \quad$$ (b) $$[10,12]$$

Step-by-Step Solution

Answer

After calculations, it was found that the average rates of change for the intervals [0, 2] and [10, 12] are \(f'(x)\) for [0,2] and \(f'(x)\) for [10,12] respectively.

Step 1: Calculate the function values

Determine the values of the function \(f(x)=\sqrt{4x+1}\) at the endpoints of each interval. For the interval [0, 2], these are \(f(0)=\sqrt{4*0+1}\) and \(f(2)=\sqrt{4*2+1}\). Similarly, for the interval [10, 12], these are \(f(10)=\sqrt{4*10+1}\) and \(f(12)=\sqrt{4*12+1}\).

Step 2: Substitute the values into the formula

Now that you have the values of \(f(0)\), \(f(2)\), \(f(10)\) and \(f(12)\), substitute these into the formula for the average rate of change, which is \((f(b) - f(a)) / (b - a)\). For the interval [0, 2], the calculation is \((f(2) - f(0)) / (2 - 0)\), and for the interval [10, 12], the calculation is \((f(12) - f(10)) / (12 - 10)\).

Step 3: Evaluate the results

After substituting and simplifying, the average rates of change for the intervals [0, 2] and [10, 12] are found. Thus the solutions have been computed.

Key Concepts

Function Evaluation

When we talk about function evaluation, we're simply plugging numbers into a function to see what they equal. In this context, we use the function evaluation to find out what the function's outputs are at specific points.
For the function given, which is \( f(x) = \sqrt{4x + 1} \), determining the values at specific points in an interval is necessary. For instance:

• To evaluate at \( x = 0 \), plug \( 0 \) into the function: \( f(0) = \sqrt{4 \times 0 + 1} = \sqrt{1} = 1 \).
• To evaluate at \( x = 2 \), plug \( 2 \) into the function: \( f(2) = \sqrt{4 \times 2 + 1} = \sqrt{9} = 3 \).

These values are essential later when calculating rates of change. It's like figuring out how one variable (\(x\)) affects another (the output of the function).
Function evaluation is a critical step because it lays the groundwork for more complex mathematical operations like finding the average rate of change.

Interval Calculations

Interval calculations involve analyzing how changes in an interval affect the function. For average rate of change, it's like finding the "speed" of the function's growth over that span. Imagine you're watching how fast something is moving over specific periods.
To do this efficiently, after evaluating the function at both ends of the interval, you utilize the formula:

• \[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]

This formula helps you see how much the function's output changes per unit interval, resembling the concept of slope in linear functions. For calculating, say from \([0, 2]\):

• \[ \frac{f(2) - f(0)}{2 - 0} = \frac{3 - 1}{2} = 1 \]

And from \([10, 12]\):

• \[ \frac{f(12) - f(10)}{12 - 10} = \frac{7 - 6}{2} = 0.5 \]

These calculations show how effectively the function changes between selected points, offering insights into function behavior over different ranges.

Square Root Function

Square root functions are a type of radical function and have a unique structure and behavior. They take the form \( f(x) = \sqrt{x} \), where the output is derived by finding the square root of \( x \). For the function in question, \( f(x) = \sqrt{4x + 1} \), it's slightly more complex due to the linear transformation inside the square root.

• This function has what's called a domain, defined by all \( x \) values that keep the expression under the radical non-negative. For \( \sqrt{4x + 1} \), start looking where \( 4x + 1 \geq 0 \).
• The solution is \( x \geq -\frac{1}{4} \), meaning any \( x \) greater than \(-\frac{1}{4}\) is valid for this function.

Additionally, these functions have a characteristic curve. They start slow and then rise more steeply as \( x \) increases, never dipping below the x-axis. The behavior changes, such as increasing intervals, become clear when analyzing rates of change in different parts of the domain.