Problem 72 - Calculus and its applications

Question

For each function, find the points on the graph at which the tangent line is horizontal. If none exist, state that fact. $$ y=4 $$

Step-by-Step Solution

Answer

The tangent line is horizontal at every point on the graph.

Step 1: Understand the Problem

The problem asks us to find points on the graph of the function where the tangent line is horizontal. For a function, a horizontal tangent line occurs at points where the derivative of the function is zero.

Step 2: Differentiate the Function

The function given is a constant function: \[ y = 4 \] The derivative of a constant function is zero: \[ \frac{dy}{dx} = 0 \]

Step 3: Analyze the Derivative

Since the derivative of the function \( \frac{dy}{dx} \) is zero everywhere, this means that the tangent line is horizontal at every point on the graph of the function.

Step 4: Solution Interpretation

With a horizontal tangent everywhere, the graph itself is a horizontal line parallel to the x-axis with a constant value of 4. Therefore, every point on this line has a horizontal tangent. The graph has no restrictions.

Key Concepts

Constant Function

A constant function is a type of mathematical function where the output value remains the same regardless of the input. Essentially, its value doesn't change as the variable changes. In symbolic terms, it can be represented as

• \( y = c \), where \( c \) is a constant.

For example, in the function \( y = 4 \), the value of \( y \) is always 4, no matter what value \( x \) takes. The graph of such a function is a straight, horizontal line. This line runs parallel to the x-axis and lies above the x-axis at the value of that constant.
The simplicity of the constant function makes it especially easy to analyze, as its derivative calculation is straightforward. There are no changes or slopes to worry about because everything stays the same across all x-values.

Derivative

The derivative of a function represents how that function changes as its input changes. For a constant function, where the output is static regardless of input, the derivative is especially straightforward. The general rule:

• For any constant \( c \), \( \frac{d}{dx}(c) = 0 \).

Why is this? Because the value of the function isn’t changing, so the 'rate of change' is zero. This zero derivative confirms that there’s no slope to calculate - the function line is flat. This makes the entire graph of a constant function have a horizontal tangent line. In other words, anywhere you touch this line, the slope remains at zero. This is a crucial insight for understanding graph behaviors in calculus: where there is no change, there is no slope.

Graph Analysis

Graph analysis involves inspecting the visual representation of a function to understand its properties better. For constant functions like \( y = 4 \), graph analysis simplifies significantly:

• The graph is a horizontal line at \( y = 4 \).
• This line extends endlessly in both directions along the x-axis.
• The height of the line (the y-value) never changes.
• All points on this line have a horizontal tangent because the slope is zero throughout.

Such graphs emphasize stability and lack of change, showcasing a visual demonstration of the derivative being zero. It's an excellent example to start with when learning about derivatives, as it visually reinforces the idea that a constant input leads to zero change. Understanding these graphs helps learners see how functions behave linearly and is key in observing more complex behavior in non-linear functions.