Problem 71 - Calculus and its applications

Question

Is the function given by \(g(x)=4 x^{3}-6 x\) continuous on \(\mathbb{R} ?\)

Step-by-Step Solution

Answer

The function is continuous on \(\mathbb{R}\) because it is a polynomial.

Step 1: Understand the type of function

The function given is a polynomial: \(g(x) = 4x^3 - 6x\). Polynomials are expressed as sums of powers of \(x\) multiplied by coefficients.

Step 2: Determine properties of polynomial functions

Polynomials are continuous functions. They do not have breaks, jumps, or points of discontinuity. This is because they are defined for all real numbers (\(\mathbb{R}\)) and can be differentiated and integrated over the entire real number line.

Step 3: Conclusion about continuity on \(\mathbb{R}\)

Since \(g(x)=4x^3-6x\) is a polynomial function, it is continuous for all \(x\in \mathbb{R}\). There are no values of \(x\) at which the function is undefined, so \(g(x)\) is continuous everywhere.

Key Concepts

Continuity in Calculus

In calculus, continuity is a fundamental concept that describes whether a function is smooth and unbroken over its domain. A function is deemed continuous if, intuitively, you can draw its graph without lifting your pen from the paper. More formally, a function \( f(x) \) is continuous at a point \( c \) if the following three conditions hold:

• \( f(c) \) is defined: This means the function has a value at \( c \).
• \( \lim_{{x \to c}} f(x) \) exists: The limit of the function as it approaches \( c \) from either side must exist.
• \( \lim_{{x \to c}} f(x) = f(c) \): The value of the function and the limit as \( x \) approaches \( c \) must be the same.

Polynomials are a key example of continuous functions. They satisfy all these conditions over their entire domain of real numbers. This makes them an essential type of function in both mathematics and its applications.

Real Numbers in Mathematics

Real numbers, symbolized by \( \mathbb{R} \), form the set of all numbers that can represent a point on an infinitely long line. This set includes a variety of numbers:

• Integers: Whole numbers that can be positive, negative, or zero.
• Fractions: Ratios of integers.
• Decimals: Numbers with a decimal point that may terminate or repeat.
• Irrational Numbers: Numbers that cannot be expressed precisely as a fraction, such as \( \pi \) and \( \sqrt{2} \).

The real number line is crucial in many areas of mathematics, including calculus. Real numbers serve as the domain for polynomial functions, among others, which are continuous and differentiable for every real number. The exploration of real numbers allows us to discuss continuity and differentiability, key topics in calculus.

Properties of Polynomial Functions

Polynomial functions possess several distinctive properties that make them particularly useful in mathematical analysis and applications. Here are some key properties:

• Continuous: As mentioned, polynomial functions are always continuous across their domain, which is the set of all real numbers \( \mathbb{R} \).
• Differentiable: These functions can be differentiated at every point in their domain. This means you can find their rate of change, or slope, at any point.
• Smoothness: Polynomial functions do not have sharp corners or cusps but are smooth curves, making them ideal for modeling real-world phenomena.
• Bounded behavior: Depending on the degree of the polynomial, the function may approach infinity or negative infinity but always following a predictable pattern of growth.

These qualities make polynomial functions extremely versatile in calculus and algebra, facilitating the study and application of continuous and differentiable functions.