Problem 32 - Calculus Early Transcendentals

Question

\(29-38\) Determine the set of points at which the function is continuous. $$F(x, y)=e^{x^{2} y}+\sqrt{x+y^{2}}$$

Step-by-Step Solution

Answer

The function is continuous for points \( (x,y) \) where \( x \geq -y^2 \).

Step 1: Analyze Continuity of Exponential Function

The first term of the function is \( e^{x^2 y} \). For exponentiation functions, \( e^{g(x,y)} \) is continuous everywhere provided that \( g(x,y) \) is continuous. Here, \( x^2 y \) is a polynomial and thus continuous everywhere in \( \mathbb{R}^2 \). Therefore, \( e^{x^2 y} \) is continuous for all \( (x, y) \).

Step 2: Analyze Continuity of Square Root Function

The second part of the function is \( \sqrt{x + y^2} \). The square root function \( \sqrt{u} \) is continuous for \( u \geq 0 \). Hence, \( x + y^2 \geq 0 \) is required for continuity, implying \( x \geq -y^2 \). This means the function \( \sqrt{x + y^2} \) is continuous at points in \( \mathbb{R}^2 \) where \( x \geq -y^2 \).

Step 3: Combine Conditions for Continuity

The entire function \( F(x, y) = e^{x^2 y} + \sqrt{x + y^2} \) is continuous wherever both components are continuous. From the previous steps, \( F(x, y) \) is continuous for all points \( (x, y) \) such that \( x \geq -y^2 \).

Step 4: Conclusion

The set of points where \( F(x, y) \) is continuous is the region in \( \mathbb{R}^2 \) defined by \( x \geq -y^2 \). Therefore, the function is continuous for all points above or on the parabola \( x = -y^2 \).

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form \( e^{u} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. These functions are crucial across various calculus problems due to their unique properties that involve continuity and differentiability.Key Properties of Exponential Functions:

• Continuous Everywhere: Exponential functions remain continuous over the entire real number line, \( \mathbb{R} \), as long as the exponent \( u \) is composed of continuous functions.
• Growth Rate: They exhibit a rapid growth or decay rate, depending on the sign of the exponent. If the exponent is positive, the function grows exponentially; if negative, it decays.

When considering a composite function like \( e^{x^2 y} \), the function remains continuous over all real numbers \((x, y)\) because \( x^2 y \) is a polynomial, and polynomials are themselves continuous over \( \mathbb{R}^2 \). This property ensures \( e^{x^2 y} \) does not introduce any discontinuity into the overall function \( F(x, y) \).

Polynomial Continuity

Polynomials are among the simplest and most straightforward functions in calculus due to their inherent continuous nature. With no gaps, jumps, or sudden changes in value, they are predictable and stable.Characteristics of Polynomial Functions:

• Smooth and Unbroken: Polynomial functions are smooth lines or curves that extend infinitely without breaks.
• Defined for All Real Numbers: They are wholly defined for every point in \( \mathbb{R} \), without restrictions, making them exclusively continuous over their entire domain.

For example, the polynomial term \( x^2 y \) from our function \( F(x, y) \) is continuous because both \( x^2 \) and \( y \) are simple polynomials. This contributes to the continuity of any composite function it forms a part of. In this case, it affirms the continuity of the exponential component \( e^{x^2 y} \).

Square Root Function

The square root function, symbolized as \( \sqrt{x} \), is inherently continuous wherever defined, though it requires specific conditions to maintain this continuity.Understanding the Continuity of Square Root Functions:

• Domain Restriction: The square root function is continuous as long as its argument is non-negative, i.e., \( u \geq 0 \).
• Graph Characteristics: The graph of \( \sqrt{x} \) is a curve that starts at the origin and extends to the right, reflecting the non-negative results of square roots in real numbers.

Considering the function \( \sqrt{x + y^2} \), continuity occurs where \( x + y^2 \geq 0 \). This condition means our function is defined and behaves predictably only where \( x \geq -y^2 \). Therefore, the region in \( \mathbb{R}^2 \) where \( F(x, y) \) remains continuous is above or on the parabola defined by \( x = -y^2 \). This ensures the overall continuity of the composite function \( F(x, y) = e^{x^2 y} + \sqrt{x+y^2} \).