Problem 4 - Calculus Concepts

Question

For Activities 1 through \(6, \quad\) for each linear model a. give the slope of the line defined by the equation. b. write the rate of change of the function in a sentence of interpretation. c. evaluate and give a sentence of interpretation for \(f(0)\). The quantity of tomatoes harvested is \(f(x)=5 x+6\) hundred pounds when \(x\) inches of rain fall.

Step-by-Step Solution

Answer

The slope is 5. Each inch of rain increases the harvest by 500 pounds. Without rain, 600 pounds are harvested.

Step 1: Identify the Slope

The slope of the linear function can be identified directly from the equation in the standard form, which is generally written as \(f(x) = mx + b\), where \(m\) represents the slope. In the function \(f(x) = 5x + 6\), the slope \(m\) is \(5\).

Step 2: Interpret the Rate of Change

The slope, \(5\), represents the rate of change of the function. This means that for every increase of 1 inch of rainfall, the quantity of tomatoes harvested increases by \(5\) hundred pounds. This is because the slope describes how much the output (tomato quantity) changes with a change in input (rainfall).

Step 3: Evaluate \(f(0)\)

To find \(f(0)\), substitute \(x = 0\) into the function: \(f(0) = 5(0) + 6 = 6\).

Step 4: Interpret \(f(0)\)

The value of \(f(0)\) is \(6\), representing the base quantity of tomatoes harvested when there is no rainfall. This means that even without any rain, 600 pounds (hundreds x 100) of tomatoes are harvested.

Key Concepts

Understanding the Slope in Linear Functions

The slope of a linear function is a fundamental concept that tells us how steep a line is. It's what links the input (independent variable) to the output (dependent variable). In mathematical terms, the slope is denoted as \(m\) in the linear equation \(f(x) = mx + b\). This slope \(m\) indicates how much \(f(x)\) (output) changes for a one-unit change in \(x\) (input).

For example, in the function \(f(x) = 5x + 6\), the slope \(m\) is \(5\). This means, quite simply, every time \(x\) increases by \(1\), \(f(x)\) increases by \(5\).

• Positive slope: Line rises as it moves from left to right.
• Negative slope: Line falls as it moves from left to right.
• Zero slope: A horizontal line with no rise.

Understanding the slope helps you quickly assess how an increase or decrease in one variable will influence the other in a linear relationship.

Decoding the Rate of Change

The rate of change in a linear function provides a real-world interpretation of the slope. It's notably the change in the dependent variable (output) corresponding to a change in the independent variable (input).

Taking our earlier example \(f(x) = 5x + 6\), the rate of change is \(5\). In practical terms, this means for every inch of rainfall, the number of harvested tomatoes rises by \(5\) hundred pounds. It's essential to understand that the rate of change is continuous, implying it reflects any small change of \(x\).

• Helps predict outcomes: Quantifies the effect of increasing or decreasing \(x\).
• Informs decision-making: Useful in contexts where knowing how outputs react to changes in inputs is crucial, like business and economics.

Thus, the rate of change is more than just a number—it's an insight into how the relationship between two quantities progresses.

Mastering Function Evaluation

Function evaluation is the process of determining the value of a function given a specific input. It's a way to explore how the output changes as inputs vary.

For instance, if we evaluate the function \(f(x) = 5x + 6\) at \(x = 0\), we compute the value of \(f(0)\). By substituting \(0\) for \(x\), we calculate \(f(0) = 5(0) + 6 = 6\). This gives us the base output or starting point, which often holds significance in real-life scenarios.

• At \(x = 0\): Often represents a starting condition or baseline measurement.
• Provides concrete values: Helps visualize outcomes at specific instances.

In our scenario, \(f(0) = 6\) dictates that when no rain falls, the base harvest of tomatoes is \(600\) pounds. Function evaluation hence aids in understanding crucial points like selling prices, costs, or, as in our example, crop yields in the absence of varying factors.