Problem 4 - Calculus for Biology and Medicine

Question

Find the derivative at the indicated point from the graph of each function. $$ f(x)=-5 x+1 ; x=0 $$

Step-by-Step Solution

Answer

The derivative of the function at \( x = 0 \) is \( -5 \).

Step 1: Understand the Function

The function given is a linear function: \( f(x) = -5x + 1 \). Linear functions have a constant derivative, which is the slope of the line.

Step 2: Identify the Slope

For the function \( f(x) = -5x + 1 \), the coefficient of \( x \), which is \( -5 \), represents the slope of the line. Hence, the derivative of the function is \( f'(x) = -5 \).

Step 3: Evaluate the Derivative at the Point

Since the derivative of the function is constant, \( f'(x) = -5 \) at all points. Thus, at \( x = 0 \), the derivative is \( f'(0) = -5 \).

Step 4: Conclusion

The derivative at the indicated point, \( x = 0 \), is simply the slope of the function, which is \( -5 \).

Key Concepts

Derivative

When dealing with calculus, a core concept is the derivative, which represents how a function changes as its input changes. For any real-valued function \( f(x) \), the derivative \( f'(x) \) symbolizes the function's rate of change or the gradient of the tangent line at any specific point.

• The derivative helps us understand the function's behavior, such as where it is increasing or decreasing.
• If the function is a straight line, as in a linear function, the derivative is constant.
• Derivatives allow us to perform more advanced calculus operations like finding maxima and minima of functions.

In our example, the function provided is linear; hence, finding its derivative is simpler, without the need for complex calculations usually involved with non-linear functions.

Linear Function

A linear function is one of the simplest types of functions in mathematics. It is often written in the form \( f(x) = mx + b \) where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, which indicates where the line crosses the y-axis.

These functions are straight lines when graphed on a coordinate plane. They have unique characteristics:

• Their graphs are symmetric around a single line.
• Their derivatives are constant, as the rate of change is the same throughout the function.
• They model a direct relationship between the two variables involved.

In the exercise, the function \( f(x) = -5x + 1 \) is linear with a negative slope, indicating that the line is falling as we move from left to right on the graph.

Slope

The slope of a line is a measure of its steepness and direction. In the context of linear functions, the slope is a crucial component that defines the line's incline:

• It corresponds to the coefficient of \( x \) in the line's equation.
• A positive slope means the line rises, while a negative slope signifies the line descends.
• The greater the absolute value of the slope, the steeper the line.

For the function \( f(x) = -5x + 1 \), the slope \( m \) is \(-5\). This reveals that the line is falling at a constant rate as we progress from left to right on the graph, moving five units down vertically for every unit moved to the right.

Function Evaluation

Function evaluation involves substituting a specific input into the function to find the output. This process helps us understand how the function behaves at particular points.

• We plug in a value for \( x \) to find the corresponding \( f(x) \).
• This process is straightforward for linear functions, but it becomes more complex with non-linear counterparts.
• The same concept applies when evaluating the derivative of a function at a certain point.

In this exercise, evaluating the derivative of the function \( f(x) = -5x + 1 \) at \( x = 0 \) is simple due to the constant nature of the derivative for linear functions. Since the slope is constant, the derivative at any point, including \( x=0 \), is \(-5\). This demonstrates the straightforward nature of linear functions and their derivatives.