Problem 71 - 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. $$ y=-3 $$

Step-by-Step Solution

Answer

Every point on the graph has a horizontal tangent line.

Step 1: Understanding the Function

The given function is a constant function, which means the value of the output, or $y$, does not change as the input $x$ changes.

Step 2: Identifying the Derivative

For constant functions, the derivative with respect to \(x\) is 0. Therefore, the derivative of \(y = -3\) with respect to \(x\) is 0, or \( \frac{dy}{dx} = 0 \).

Step 3: Interpreting the Derivative

A horizontal tangent line occurs where the derivative is 0. Since the derivative is 0 for this function at all points, the tangent line is horizontal everywhere on the graph of the function.

Step 4: State the Result

Thus, every point on the graph of the function \( y = -3 \) is a point at which the tangent line is horizontal.

Key Concepts

Constant Function

A constant function is a unique type of function where the output value remains the same no matter what input value is given. In mathematical terms, it is written as \( y = c \), where \( c \) is a constant number. For example, the function \( y = -3 \) is a constant function because \( y \) will always equal \(-3\) regardless of the \( x \) value.
A useful way to think about constant functions is to imagine a perfectly flat, horizontal line on a graph that never rises or dips. This straight line implies that no matter how far you move along the x-axis, the y-value does not change. Because of this characteristic, constant functions are considered the simplest type of functions in calculus.
Constant functions play a crucial role in understanding calculus concepts because they set a foundation for more complex calculations.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. In simpler terms, it tells you the slope or the steepness of the tangent line at any given point on a function.
When dealing with a constant function, such as \( y = -3 \), its derivative with respect to \( x \) is always zero. In the notation, this is written as \( \frac{dy}{dx} = 0 \). This zero derivative means there is no change in the y-value as the x-value changes, reflecting the flat nature of constant functions.

• Zero Derivative: Indicates a flat slope.
• Flat Slope: Corresponds to a horizontal line.

This property of having a zero derivative at all points is why constant functions are simpler compared to other types of functions that have varying slopes.

Tangent Line

The tangent line at any particular point on a curve is a straight line that "just touches" the curve at that single point. The slope of this tangent line is determined by the derivative at that point. For most functions, the slope of the tangent line changes depending on which point it is touching. However, in the case of a constant function, the situation is quite different.
Since the derivative of a constant function like \( y = -3 \) is zero, the tangent line is horizontal at every point on the graph. This means that anywhere you choose a point on the graph, the tangent line will be completely flat. Such a line doesn’t angle upwards or downwards, it remains level and parallel to the x-axis.

Graph of a Function

A graph of a function is a visual representation of how the values of the function behave when plotted on a coordinate system. In the case of constant functions, their graphs form simple horizontal lines across the coordinate plane.
For the function \( y = -3 \), the graph is a horizontal line that cuts across the y-plane at \( y = -3 \), parallel to the x-axis. Because every point on this graph is the same, it makes interpreting the behavior of the constant function extremely easy.

• Horizontal Line: Represents the unchanging y-value.
• Parallel to X-axis: Shows uniformity and stability.

Understanding the graph of a function is crucial because it allows you to see the relationship between variables in function form visually.