Question

Determining if a Function Is Homogeneous In Exercises \(67-74,\) determine whether the function is homogeneous, and if it is, determine its degree. A function \(f(x, y)\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y)\) $$ f(x, y)=\tan (x+y) $$

Short answer

The function \(f(x, y)=\tan (x+y)\) is not homogeneous.

Step-by-step solution

Substitute \(x, y\) with \(tx, ty\) in the function

Calculate \(f(tx, ty)\) by substituting \(x, y\) with \(tx, ty\) in the function \(f(x, y)=\tan(x+y)\). This gives us \(f(tx, ty)=\tan(tx+ty)\).

Analyze the new function

Observe that the new function cannot be represented as \(t^{n} \tan (x+y)\) for any positive integer \(n\). This is because \(t\) is being added inside the tangent function, not multiplied outside the function.

Conclusion

Therefore, the function \(f(x, y)=\tan (x+y)\) is not homogeneous as it does not satisfy the condition \(f(t x, t y)=t^{n} f(x, y)\) for any positive integer \(n\).

Key concepts

Degree of Homogeneity

Understanding the degree of homogeneity is essential when analyzing functions in various branches of mathematics, including calculus and algebra. The degree of homogeneity refers to a property of a function that describes how the values of the function scale as its variables are proportionally scaled. This concept is pivotal when analyzing the behavior of functions in both theoretical and applied contexts.

Consider a function of two variables, denoted as f(x, y). This function is termed homogeneous of degree n if, when we scale both variables by the same factor t, the function's value is multiplied by t^n. Mathematically, this condition is written as f(tx, ty) = t^n f(x, y). The integer n here represents the degree of homogeneity.

Applying the Concept to tan(x + y)

In the given exercise, determining if the function f(x, y) = tan(x + y) is homogeneous involves substituting x and y with tx and ty respectively, and then seeing if the function scales according to the definition. As we see from the solution, the function does not satisfy the condition for any positive integer n, thus it is not homogeneous.

Calculus

Calculus, an indispensable tool in mathematics and physics, comprises two main branches: differential calculus and integral calculus. Differential calculus focuses on the concept of the derivative, which measures how a function changes as its input changes. Conversely, integral calculus focuses on the concept of the integral, which represents accumulation or the area under a curve.

In the context of the homogeneous functions, differentiability plays a significant role. If a function is homogeneous, it can simplify the process of differentiating it by reducing the complexity of the function's expression, especially when applying the chain rule. In the context of our exercise involving the tangent function, we assess differentiability through its homogeneity.

Homogeneity and Differentiation

When evaluating if f(x, y) = tan(x + y) is homogeneous and, by extension, exploring its differential properties, we discovered that the function does not simplify as required for homogeneous functions. As a result, its differentiation requires using the standard rules of derivatives without simplifications that the property of homogeneity might provide.

Tangent Function

The tangent function, commonly denoted as tan, is a fundamental trigonometric function that relates an angle of a right-angled triangle to the ratio of the opposite side to the adjacent side. In more advanced mathematics, the tangent function is explored in the context of periodicity, asymptotic behavior, and its relationship with other trigonometric functions.

However, one less commonly discussed aspect of the tangent function is how it behaves under scaling, an important aspect when considering whether a function is homogeneous. The key property to remember is that the tangent of a sum, tan(x + y), is not the same as the sum of tangents (tan(x) + tan(y)).

Non-Homogeneous Behavior of Tangent

In our specific exercise, we are applying the concept of homogeneity to f(x, y) = tan(x + y). If the function were homogeneous, there would be a consistent way to factor out the scaling variable t from the function. However, since the angle sum in the tangent function does not allow for simple multiplication by t^n outside the function, it breaks the homogeneity rule leading to the conclusion that the tangent function within our exercise is non-homogeneous.