Problem 77 - Applied Calculus

Question

If a linear function passes through two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) what is the average value of the function on the interval from \(x_{1}\) to \(x_{2}\) ?

Step-by-Step Solution

Answer

The average value of the linear function over the interval is \(\frac{y_1 + y_2}{2}\).

Step 1: Identify the Points

Identify the two points given in the problem: Point 1: \((x_1, y_1)\)Point 2: \((x_2, y_2)\)

Step 2: Formula for Average Value

The average value of a function \(f\) on an interval \([a, b]\) is given by \(\frac{1}{b-a} \int_{a}^{b} f(x) \, dx\). In this case, the interval is \([x_1, x_2]\).

Step 3: Equation of the Line

Find the equation of the line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\). Using the slope formula, the slope \(m\) is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]The equation of the line is:\[ y = m(x - x_1) + y_1 \]

Step 4: Write the Linear Function

Substituting the slope into the linear equation:\[ y = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) + y_1 \]This gives the function \(f(x)\) that needs to be averaged over \([x_1, x_2]\).

Step 5: Integrate the Function

Integrate the function \(f(x) = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) + y_1\) over the interval \([x_1, x_2]\):\[ \int_{x_1}^{x_2} \left( \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) + y_1 \right) \, dx \]

Step 6: Calculate the Definite Integral

Carry out the integration. The integral of a linear function \((ax + b)\) over \([x_1, x_2]\) can be found as:\[ \int (\frac{y_2 - y_1}{x_2 - x_1} x + C) \, dx = \frac{y_2 - y_1}{x_2 - x_1} \frac{x^2}{2} + Cx \]Compute this from \(x_1\) to \(x_2\).

Step 7: Calculate Average Value

Use the formula for the average value:\[ \frac{1}{x_2 - x_1} \left[ \int_{x_1}^{x_2} \left( \frac{y_2 - y_1}{x_2 - x_1} \right) (x - x_1) + y_1 \, dx \right] \]Simplify it to obtain the average value.

Step 8: Simplify Expression

Upon further simplification, the average value of any linear function over an interval is the midpoint of the function values at the two ends, therefore:\[ \text{Average value} = \frac{y_1 + y_2}{2} \]

Key Concepts

Linear Function

A linear function is a mathematical expression that creates a straight line when graphed on a coordinate plane. It typically follows the form \[ f(x) = mx + c \]where

• \( m \) represents the slope of the line, indicating how steep or flat the line is.
• \( c \) is the y-intercept, the point where the line crosses the y-axis.

Linear functions are simple yet very powerful for modeling relationships where changes between variables happen at a constant rate.

For example, if you are walking at a constant speed, the relationship between the time walked and the distance covered can be modeled by a linear function. This function is defined by its slope and y-intercept. Understanding how to determine these components is essential for working with linear functions.

Definite Integral

The definite integral is a mathematical concept used to find the accumulation of quantities over an interval, which in the context of a linear function, involves calculating area under the curve of the function. It is depicted as \[ \int_{a}^{b} f(x) \, dx \]where:

• \( a \) and \( b \) are the boundaries of the interval over which you're calculating the integral.
• \( f(x) \) is the function being integrated.

The definite integral has wide applications, such as finding total distance traveled given a speed-time function or the total cost in economics when dealing with marginal cost functions.

In this exercise, we use it to compute the average value of a linear function by integrating over a given interval and then dividing by the interval's length.

Slope of a Line

The slope of a line is a numerical measure of its inclination or declination, representing the rate of change of the function. For a linear function passing through two points, it is calculated as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here,

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

The slope determines how much \( y \) changes when \( x \) increases by one unit. A positive slope means the line rises as it moves from left to right, whereas a negative slope indicates the line falls.

Understanding how to find the slope is key in forming the equation of a line, as the slope is used to determine the angle or tilt of the line in relation to the axes.

Equation of a Line

The equation of a line is a way to mathematically describe every point on a linear path. By using the slope and a known point on the line, you can determine this equation. The general formula is:\[ y = mx + b \]Where,

• \( m \) is the slope, indicating how steep the line is.
• \( b \) is the y-intercept, showing where the line cuts through the y-axis.

For lines through two specific points, the equation can be written using the point-slope form:\[ y - y_1 = m(x - x_1) \]This form allows you to plug in the slope and coordinates of a point through which the line passes to derive the full equation.

The equation is fundamental when calculating the average value of a function or integrating the function over an interval.