Problem 4 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Given Function

Identify the function to work with, which is given as \(f(x)=-5x+1\). This is a linear function.

Step 2: Understand Derivative of Linear Functions

Recognize that for linear functions of the form \(f(x) = ax + b\), the derivative is simply the coefficient of \(x\), which is \(a\).

Step 3: Derive the Function

Apply the derivative to the function \(-5x+1\). Since this is a linear function, the derivative \(f'(x)\) is simply \(-5\).

Step 4: Evaluate at Given Point

Evaluate the derivative at \(x = 0\). Since the derivative of a linear function is a constant, \(f'(0)\) remains \(-5\).

Key Concepts

Linear Function

When we encounter a linear function, we are dealing with the simplest type of function, which looks like a straight line on a graph. A typical linear function has the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. This function represents a straight line where:

• \( a \) is the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

In the example given, \( f(x) = -5x + 1 \), this means that:

• \( a = -5 \): The line slopes downward, as indicated by the negative sign.
• \( b = 1 \): The line crosses the y-axis at \( y = 1 \).

Linear functions are extremely important in math and science because they make calculations simpler and help illustrate basic relationships. Since they don't curve or change direction, evaluating them and their characteristics, such as derivatives, is straightforward.

Evaluate Derivative

To evaluate the derivative of a function means to find the rate at which the function is changing at any given point. For linear functions, this process is especially simple. A derivative tells us how steep the line is at each point, which directly relates to the concept of slope.For our linear function \( f(x) = -5x + 1 \), the derivative is easy to determine, as it is the constant slope of the line. This slope, or rate of change, for all linear functions \( f(x) = ax + b \) is just \( a \). Therefore, in our function, \( a = -5 \), which means:

• \( f'(x) = -5 \): The slope of the line and the rate of change everywhere on the line.

Because the slope doesn't change in linear functions, evaluating the derivative at any point along the function, such as \( x = 0 \), results in the same value, \( -5 \), indicating the line's uniformity in slope.

Derivative Calculation

Derivative calculation involves determining the slope or rate of change of a function at any point. For our function \( f(x) = -5x + 1 \), this involves a straightforward process due to its linear nature.The steps involved in calculating the derivative of a linear function like \( f(x) = ax + b \) are:

• Identify the coefficient of \( x \), which is \( a \) in our equation. Here, it is \( -5 \).
• The derivative of a linear function is simply its slope, \( f'(x) = a \). Thus, for our function, \( f'(x) = -5 \).

By calculating the derivative, we can quickly understand and describe how the function behaves. Since linear functions have constant slopes, computing these derivatives is especially easy, making it an ideal starting point for understanding more complex differentiation concepts.