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Chapter 1: Problem 11

Find values of \(x,\) if any, at which \(f\) is not continuous. $$ f(x)=5 x^{4}-3 x+7 $$

Short Answer

Expert verified
The function \( f \) is continuous for all \( x \).

Step by step solution

01

Understand the Continuity of Polynomial Functions

Polynomial functions, like the given function \( f(x) = 5x^4 - 3x + 7 \), are continuous everywhere. This means they are continuous for all real numbers \( x \). There are no points of discontinuity in polynomial functions by definition, as they are smooth and unbroken for all values of \( x \).
02

Justify with Detailed Explanation

The given function \( f(x) = 5x^4 - 3x + 7 \) is a fourth-degree polynomial. Polynomials are defined and continuous over all real numbers. There are no breaks, jumps, or asymptotes that can interrupt the continuity of the graph of a polynomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
Polynomial functions are mathematical expressions involving a sum of powers of variables multiplied by coefficients. They have the general form: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \] Here, the highest power of the variable, known as the degree of the polynomial, determines its behavior and shape.
  • Coefficients: These are the numbers that multiply the variable's terms, like the 5, -3, and 7 in the function \( f(x) = 5x^4 - 3x + 7 \).
  • Degree: This is the highest power of \( x \) in the expression. For \( f(x) = 5x^4 - 3x + 7 \), the degree is 4 because of the term \( 5x^4 \).
  • Constant Term: This is the term without the variable, in this case, 7.
Polynomial functions are smooth and continuous across their domains, which means they have no jumps, sharp corners, or breaks.
Discontinuity
Discontinuity occurs at points where a function is not continuous. This could be due to jumps, vertical asymptotes, or removable discontinuities in a function. For a function to be continuous, it should have no sudden changes or breaks in its graph. Key characteristics of discontinuities in graphical terms include:
  • Jump Discontinuity: This happens when there is a sudden leap in function values at a point, like in piecewise functions.
  • Vertical Asymptote: This is when the function approaches infinity at a particular point, common in rational functions.
  • Hole or Removable Discontinuity: This is when a function is not defined at a point, but can be "fixed" by redefining the value, often observed in rational functions after cancellation.
For polynomial functions, however, none of these apply. Polynomial functions are inherently continuous over all real numbers, making them free from any form of discontinuity.
Real Numbers
Real numbers incorporate all possible numbers that can exist on the number line, encompassing both rational and irrational numbers. This includes integers, fractions, and numbers with decimals without any imaginary components. Highlights of real numbers:
  • Rational Numbers: These are numbers that can be expressed as a ratio of two integers, such as \( \frac{3}{4} \) or -2.
  • Irrational Numbers: These numbers cannot be expressed as simple fractions and have non-repeating, non-terminating decimal parts, like \( \pi \) or \( \sqrt{2} \).
  • Continuum: The set of real numbers forms a continuous line without gaps. This continuity is mirrored by polynomial functions, making them continuous for all real values of \( x \).
Real numbers serve as the domain for polynomial functions, ensuring smooth graph transitions without any breaks from \(-\infty \) to \(+\infty \). Every real number can be plugged into a polynomial function, yielding a corresponding output smoothly.

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Most popular questions from this chapter

Find values of \(x,\) if any, at which \(f\) is not continuous. $$ f(x)=\left\\{\begin{array}{ll}{2 x+3,} & {x \leq 4} \\ {7+\frac{16}{x},} & {x>4}\end{array}\right. $$

The notion of an asymptote can be extended to include curves as well as lines. Specifically, we say that curves \(y=f(x)\) and \(y=g(x)\) are asymptotic as \(x \rightarrow+\infty\) provided $$ \lim _{x \rightarrow+\infty}[f(x)-g(x)]=0 $$ In these exercises, determine a simpler function \(g(x)\) such that \(y=f(x)\) is asymptotic to \(y=g(x)\) as \(x \rightarrow+\infty\) or \(x \rightarrow-\infty\) Use a graphing utility to generate the graphs of \(y=f(x)\) and \(y=g(x)\) and identify all vertical asymptotes. $$ f(x)=\frac{x^{2}-2}{x-2} $$

In the circle in the accompanying figure, a central angle of measure \(\theta\) radians subtends a chord of length \(c(\theta)\) and a circular arc of length \(s(\theta) .\) Based on your intuition, what would you conjecture is the value of \(\lim _{\theta \rightarrow 0^{+}} c(\theta) / s(\theta) ?\) Verify your conjecture by computing the limit.

Find the limits. $$ \lim _{x \rightarrow+\infty} \cos \left(\frac{1}{x}\right) $$

On which of the following intervals is $$ f(x)=\frac{1}{\sqrt{x-2}} $$ continuous? $$ \begin{array}{llll}{\text { (a) }[2,+\infty)} & {\text { (b) }(-\infty,+\infty)} & {\text { (c) }(2,+\infty)} & {\text { (d) }[1,2)}\end{array} $$
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