Problem 65 - Algebra Form and Function

Question

\(a\) and \(x\) are positive. What is the effect of increasing \(a\) on the value of the expression? Does the value increase, decrease, or remain unchanged? $$ \frac{(a x)^{1 / 3}}{\sqrt{a}} $$

Step-by-Step Solution

Answer

Based on the analysis of the given expression and the calculated partial derivative, we can conclude that as the value of \(a\) increases, the value of the expression \(\frac{(a x)^{1 / 3}}{\sqrt{a}}\) will decrease.

Step 1: Write down the given expression

We are given the expression: $$ \frac{(a x)^{1 / 3}}{\sqrt{a}} $$

Step 2: Simplify the expression if possible

Using the properties of exponents, we can simplify the expression as follows: $$ \frac{(a x)^{1 / 3}}{\sqrt{a}} = \frac{a^{1/3}x^{1/3}}{a^{1/2}} = a^{\frac{1}{3}-\frac{1}{2}}x^{1/3} = a^{-1/6}x^{1/3} $$

Step 3: Take the partial derivative with respect to \(a\)

Calculate the partial derivative of the simplified expression with respect to \(a\): $$ \frac{\partial}{\partial a}(a^{-1/6}x^{1/3}) = -\frac{1}{6} a^{-1/6 - 1} x^{1/3} = -\frac{1}{6}a^{-7/6}x^{1/3} $$

Step 4: Analyze the sign of the derivative

Now we check the sign of the partial derivative we just calculated: $$ -\frac{1}{6}a^{-7/6}x^{1/3} $$ Since both \(a\) and \(x\) are positive, we have that the value of the expression \(a^{-7/6}\) is also positive, and the value of the expression \(x^{1/3}\) is also positive. Therefore, the product of these two expressions is positive as well. Multiplying by the negative coefficient \(-\frac{1}{6}\) makes the entire derivative negative. The negative partial derivative tells us that as we increase the value of \(a\), the value of the original expression will decrease.

Key Concepts

Partial Derivatives

In calculus, partial derivatives are used to understand how a function changes as one of its variables changes, while all other variables are held constant. This is particularly important for multivariable functions.
For example, if we have a function that depends on both variables \(a\) and \(x\), and we want to know how changing \(a\) affects the function, we compute the partial derivative with respect to \(a\).
In our exercise, we calculated \(\frac{\partial}{\partial a}(a^{-1/6}x^{1/3})\). This tells us the rate at which the function \(a^{-1/6}x^{1/3}\) changes with \(a\), while \(x\) remains constant.

• A negative partial derivative value implies the function decreases as \(a\) increases.
• A positive partial derivative implies the function increases.
• Partial derivatives are crucial for understanding the sensitivity of functions with respect to their variables.

Expression Simplification

Simplification involves reducing expressions to their most concise and easily comprehensible form. In mathematics, simplified expressions make complex calculations more manageable and errors less likely.
For this exercise, simplifying the original expression \(\frac{(a x)^{1 / 3}}{\sqrt{a}}\) is key to analyzing its behavior as \(a\) changes. By applying the properties of exponents, we derived \(a^{-1/6}x^{1/3}\).
Why simplify?

• Simplified expressions highlight significant aspects, like dependencies on variables, clarity, and potential behavior changes.
• They streamline subsequent calculations, like taking derivatives.

Being efficient with simplification is a skill that supports deeper understanding and facilitates more complex problem-solving tasks.

Properties of Exponents

The properties of exponents are fundamental rules that allow for the manipulation and simplification of expressions involving powers.
Key exponent properties include:

• \(a^m \cdot a^n = a^{m+n}\): This property simplifies products of like bases.
• \(\frac{a^m}{a^n} = a^{m-n}\): We used this property to simplify \(\frac{a^{1/3}}{a^{1/2}}\) to \(a^{-1/6}\) in our exercise.
• \((a^m)^n = a^{m \cdot n}\): Useful for dealing with power raised to another power.

Understanding these rules is critical as they help solve equations, conduct transformations, and are foundational in calculus work such as taking derivatives. By mastering these principles, you can handle a wide array of mathematical challenges effectively.