Problem 51 - College Algebra with Modeling and Visualization

Question

Evaluate \(f(x)\) at the given \(x\) Approximate each result to the nearest hundredth. $$ f(x)=x^{3 / 2}-x^{1 / 2}, \quad x=50 $$

Step-by-Step Solution

Answer

The approximate value of \( f(50) \) is 346.48.

Step 1: Understand the function.

The function given is \( f(x) = x^{3/2} - x^{1/2} \). This function consists of two parts: the term \( x^{3/2} \) and the term \( x^{1/2} \). We will need to calculate each term separately for the given value of \( x \).

Step 2: Evaluate the first term of the function.

Calculate \( x^{3/2} \) for \( x = 50 \). This can be rewritten as \( (x^3)^{1/2} \), and it is the same as taking the square root of \( 50^3 \). First, compute \( 50^3 = 125000 \). Then, find \( \sqrt{125000} \approx 353.55 \).

Step 3: Evaluate the second term of the function.

Now, evaluate \( x^{1/2} \) for \( x = 50 \) which is the square root of 50. It can be approximated as \( \sqrt{50} \approx 7.07 \).

Step 4: Calculate the function value.

Subtract the second term from the first term: \( f(50) = 353.55 - 7.07 \). This equals \( 346.48 \).

Step 5: Approximate the result to the nearest hundredth.

The calculated value \( 346.48 \) is already rounded to the nearest hundredth.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In our exercise, the function \(f(x) = x^{3/2} - x^{1/2}\) is an example of such an expression. The variables are represented by \(x\) and the arithmetic operations involve exponents, which are a type of operation in algebra. These expressions can be simplified and evaluated by substituting a specific numerical value for the variable, as we did with \(x = 50\).
Breaking down these expressions into calculable parts allows us to evaluate them effectively. This ability to manipulate and simplify algebraic expressions is foundational to understanding more complex mathematical problems. It is essential for solving equations, analyzing functions, and finding mathematical approximations.
To effectively manage these expressions:

• Identify each term and understand its components.
• Perform operations according to mathematical rules, such as the order of operations.
• Simplify expressions where possible, step by step, to avoid errors.

Exponents and Radicals

Exponents and radicals are essential elements in algebraic expressions. An exponent indicates how many times a number, known as the base, is multiplied by itself. In our function, both \(x^{3/2}\) and \(x^{1/2}\) involve exponents.
The expression \(x^{3/2}\) can be interpreted in different ways, such as \((x^3)^{1/2}\), which means the square root of \(x^3\), or equivalently \((x^{1/2})^3\), indicating the cube of the square root of \(x\). Similarly, \(x^{1/2}\) represents the square root of \(x\).
Handling these powers and roots is crucial for accurately evaluating expressions like \(f(x)\):

• Understand what the exponent signifies, particularly with fractions.
• Use a calculator for precise calculations, especially when values lead to non-integer results.
• Break down exponents into more manageable steps if needed, as intermediate results often facilitate easier calculations.

Mathematical Approximation

Mathematical approximation involves finding a nearly exact value when an exact number is unwieldy or difficult to calculate. This is especially useful when dealing with irrational numbers or complex calculations.
In this exercise, the terms \(x^{1/2}\) and \((x^3)^{1/2}\) lead to non-integer roots. Calculated precisely, \(x^{1/2}\) becomes somewhere between integers as 7.07, and similarly \((x^3)^{1/2}\) approx. 353.55. Approximations help to express these results in a more usable form, rather than dealing with lengthy decimals.
When approaching approximations:

• Understand the desired level of precision, whether it's the nearest hundredth or some other decimal place.
• Use tools, such as calculators, for consistency and reliability in achieving that precision.
• Consider which approximations are acceptable based on the context of the problem; sometimes close enough is perfect, other times exact numbers are necessary.