Problem 5 - Calculus with Concepts in Calculus

Question

Find the numerical value of the function at the given values of \(a\). $$ g(x)=1 /\left(2 x^{2}\right) ; a=\sqrt{2} $$

Step-by-Step Solution

Answer

The numerical value of \( g(x) \) at \( x = \sqrt{2} \) is \( \frac{1}{4} \).

Step 1: Understand the Function

We are given the function \( g(x) = \frac{1}{2x^2} \). This function is a rational function, where the polynomial \( 2x^2 \) is in the denominator.

Step 2: Substitute the Given Value

We need to find the value of \( g(x) \) at \( x = a \), where \( a = \sqrt{2} \). Substitute \( \sqrt{2} \) into the function: \( g(\sqrt{2}) = \frac{1}{2(\sqrt{2})^2} \).

Step 3: Simplify the Expression

Calculate \( (\sqrt{2})^2 \). The square of \( \sqrt{2} \) is 2. So, the expression becomes \( g(\sqrt{2}) = \frac{1}{2 imes 2} \).

Step 4: Perform Multiplication in the Denominator

Multiply the terms in the denominator: \( 2 \times 2 = 4 \). Substitute back into the expression: \( g(\sqrt{2}) = \frac{1}{4} \).

Step 5: Provide the Final Result

The value of the function \( g(x) \) at \( x = \sqrt{2} \) is \( \frac{1}{4} \).

Key Concepts

Rational Functions

A rational function is a very important concept in calculus, and it is essentially a fraction, or a ratio, of two polynomials. In simpler terms, a rational function takes the form \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials. The denominator, \( q(x) \), must not be zero because division by zero is undefined in mathematics. Ensuring that \( q(x) eq 0 \) is crucial for determining the domain of the function.
Rational functions can display a variety of behaviors as they depend heavily on the structure and degree of the polynomials involved.
Here are some points to consider when dealing with rational functions:

• Vertical asymptotes: Occur when the denominator polynomial (\( q(x) \)) approaches zero.


• Horizontal asymptotes: Can be determined by comparing the degrees of \( p(x) \) and \( q(x) \).


• Discontinuity points: These exist where the rational function is not defined or where \( q(x) = 0 \).

Since the example function we are discussing is \( g(x) = \frac{1}{2x^2} \), it is a simple rational function with the denominator \(2x^2\). The domain of this function is all real numbers except where the denominator equals zero, i.e., \( x eq 0 \).

Substitution Method

The substitution method is a straightforward technique used to evaluate functions at specific values. It involves replacing the variable (typically \(x\) or \(y\)) with a given number. This is especially useful in finding the numerical value of functions at designated points.
The steps for using the substitution method are as follows:

• Identify the variable you need to replace.
• Substitute the given value into the function wherever you find the variable.


• Carry out any arithmetic operations required to simplify the expression to a single numerical value.

In our example, we needed to find \( g(x) \) at \( x = \sqrt{2} \). Substituting \( \sqrt{2} \) into the function:\[ g(\sqrt{2}) = \frac{1}{2(\sqrt{2})^2} \]. This substitution allows us to evaluate the function precisely at \( x = \sqrt{2} \).
Substitution is an essential tool in calculus because it is often the first step in solving many different types of problems, from simple to complex.

Simplifying Expressions

In mathematics, simplifying expressions involves reducing them to their simplest form while retaining their value. This makes it easier to understand and work with the expression. Simplification often requires performing arithmetic operations and algebraic manipulations.
To simplify expressions, follow these guidelines:

• Carry out any exponentiation (in our example, \( (\sqrt{2})^2 \)).
• Apply operations like multiplication or division as needed (here, multiplying \(2 \times 2\)).


• Reduce fractions to their simplest form if possible.

The core idea is to manipulate the expression so that it is easy to interpret and calculate. In the example, after substituting \( \sqrt{2} \) into the function, we obtained \( \frac{1}{2 \times 2} \), which simplifies to \( \frac{1}{4} \). This process of simplification makes solving the problem clearer and the result immediate.
Simplifying expressions is not only crucial for cleaner forms of mathematical expressions but also helps in clear communication of the result.