Problem 1 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The derivative of the function at \( x = 1 \) is 0.

Step 1: Understand the Function

The function given is \( f(x) = 5 \). This is a horizontal line since the value of \( f(x) \) is constant for any \( x \). A horizontal line parallel to the x-axis results when a function is equal to a constant.

Step 2: Determine the General Derivative

The derivative of a constant function is zero, because a constant function has no rate of change as \( x \) changes. Thus, the derivative \( f'(x) \) is computed as \( \frac{d}{dx}(5) = 0 \).

Step 3: Evaluate at the Indicated Point

Although derivatives typically vary at different points, for a constant function, the derivative is the same everywhere. Therefore, at \( x = 1 \), the derivative remains \( f'(1) = 0 \).

Key Concepts

Constant Function

A constant function is a fundamental concept in calculus and algebra. A constant function, such as \( f(x) = 5 \), means exactly what its name suggests: the function remains "constant" regardless of the input \( x \). This occurs because the output value is fixed, no matter what value \( x \) takes on.
Such functions are mathematically simple as:

• Every point on the graph of a constant function lies on a horizontal line.
• The value of the function is the same for any \( x \) value.

Constant functions are represented as horizontal lines on a graph. In terms of calculus, the lack of change in the output as \( x \) varies means that the slope of this line is zero. This zero slope leads us to our next topic, where we explore more about horizontal lines and their characteristics.

Horizontal Line

The graph of a constant function is represented by a horizontal line. For example, the graph of \( f(x) = 5 \) is a line that runs parallel to the x-axis at y = 5. Horizontal lines have some interesting properties:

• They intersect the vertical axis at the value of the constant (in our case, y = 5).
• The slope of a horizontal line is zero, as there is no vertical change when \( x \) changes.

This zero slope indicates a lack of change, emphasizing that as \( x \) changes, the y-value remains perfectly still. Let's use the derivative to describe this concept further.

Rate of Change

The rate of change is an essential aspect of calculus, often described via the concept of a derivative. When analyzing how a function changes, the derivative tells us how quickly or slowly these changes occur. However, with constant functions, the rate of change is always zero.For a constant function like \( f(x) = 5 \), the derivative \( f'(x) \) is expressed as zero. This number indicates there is no change in the function's value no matter how \( x \) shifts. Therefore:

• The function maintains a steady output go across the x-axis.
• The derivative's value of zero at any \( x \) highlights the absence of any slope or incline.

This zero rate of change simplifies many analytic tasks in calculus, as the behavior of constant functions remains the same at any point, making them predictable and easy to work with.