Problem 49 - Thomas Calculus

Question

Does the curve \(y=x^{3}\) ever have a negative slope? If so, where? Give reasons for your answer.

Step-by-Step Solution

Answer

No, the curve \( y = x^3 \) never has a negative slope.

Step 1: Understand the Problem

We want to determine where the curve \( y = x^3 \) has a negative slope. To do this, we need to analyze the first derivative of the function, which represents the slope of the tangent line to the curve.

Step 2: Differentiate the Function

Find the first derivative of the function \( y = x^3 \). The derivative, \( y' \), represents the slope of the tangent line:\[ y' = \frac{d}{dx}(x^3) = 3x^2 \]

Step 3: Analyze the Derivative

The derivative \( 3x^2 \) is a quadratic function that opens upwards, as indicated by the positive coefficient of \( x^2 \). This means that \( 3x^2 \geq 0 \) for all real numbers, hence it never becomes negative.

Step 4: Conclusion from Analysis

Since \( 3x^2 \geq 0 \) for all \( x \), the slope of the curve \( y = x^3 \) is always non-negative. Therefore, the curve never has a negative slope.

Key Concepts

Derivative

In calculus, the derivative is a core concept that represents the rate at which a function is changing at any given point. Essentially, it describes the behavior of a function as you move along its curve. For a function like \( y = x^3 \), the derivative reveals the instantaneous slope of the tangent line at any given point on the curve. This helps us understand how steep or flat the curve is at a particular spot.To find the derivative of \( y = x^3 \), we compute \( y' = \frac{d}{dx}(x^3) = 3x^2 \). This resulting expression, \( 3x^2 \), tells us that as \( x \) changes, the slope of the tangent line also changes. By analyzing derivatives, we can gain insights into the decreasing or increasing nature of the function, which is crucial for understanding curve behavior.

Slope analysis

Slope analysis is an essential process in calculus, particularly when you want to gauge how a curve behaves between different points. The slope of a function at a given point is determined by its derivative. For \( y = x^3 \), we've found that the derivative is \( 3x^2 \). Since \( 3x^2 \) is always greater than or equal to zero, this indicates that the slope is never negative. This means that the curve never goes down but either stays flat or ascends as \( x \) increases.

• If the derivative is positive, the function is increasing.
• If the derivative is zero, the function is momentarily flat.

By using this analysis, we can draw conclusions about the overall trend of the function and confidently say that \( y = x^3 \) has no sections where it dips or slopes downward.

Quadratic function

A quadratic function is a polynomial function with a degree of 2, looking very much like \( ax^2 + bx + c \). In the context of derivatives, even though the original function could be of higher order, the derivative might simplify to a quadratic. For \( y = x^3 \), the derivative is \( y' = 3x^2 \), a quadratic function. The coefficient \( 3 \ (a=3) \) is positive, which means the parabola opens upward. Thus, it does not cross the x-axis and is always positive or zero.Understanding the sign and form of quadratic functions like \( 3x^2 \) helps determine key characteristics of the curve such as where it might change direction or achieve maximum or minimum slopes. This knowledge influences how we interpret the shape of the curve \( y = x^3 \).

Curve behavior

Curve behavior refers to how a graph behaves along its domain, influenced by its slope, points of inflection, and ends. The curve for \( y = x^3 \) is quite unique because its shape reveals a lot about its slope and direction. Since \( y' = 3x^2 \) never renders a negative value, the curve never decreases in value; it consistently either rises or remains flat if the power's contribution nullifies it.

• As \( x \) approaches negative infinity, the curve dips towards negative infinity itself, beginning in the third quadrant.
• As \( x \) approaches positive infinity, the curve rises towards positive infinity, moving into the first quadrant.

This understanding of curve behavior demystifies function behavior over specific intervals and tells us why \( y = x^3 \) never has a negative slope across its entirety.