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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.58
Chapter 4, Problem 4.58

Sketch the graphs of the rational functions in Exercises 53–60.


𝓍⁴ ― 1
y = ------------------
𝓍²

Verified step by step guidance
1
Step 1: Identify the rational function. The given function is \( y = \frac{x^4 - 1}{x^2} \). This can be simplified by dividing each term in the numerator by \( x^2 \).
Step 2: Simplify the function. The expression \( x^4 - 1 \) can be rewritten as \( (x^2)^2 - 1 \), which is a difference of squares. Factor it as \( (x^2 - 1)(x^2 + 1) \). Then, divide each term by \( x^2 \) to get \( y = x^2 - \frac{1}{x^2} \).
Step 3: Determine the domain. The function is undefined where the denominator is zero, so \( x \neq 0 \). The domain is all real numbers except \( x = 0 \).
Step 4: Analyze the behavior at critical points. Find the x-intercepts by setting the numerator equal to zero: \( x^4 - 1 = 0 \), which gives \( x = \pm 1 \). Check for vertical asymptotes at \( x = 0 \) and horizontal asymptotes by considering the end behavior as \( x \to \pm \infty \).
Step 5: Sketch the graph. Plot the intercepts and asymptotes, and consider the behavior of the function in different intervals. For \( x > 0 \) and \( x < 0 \), analyze the sign of the function and its concavity to sketch the graph accurately.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Functions

A rational function is a ratio of two polynomials. The function y = (x⁴ - 1) / x² is a rational function where the numerator is a polynomial of degree 4 and the denominator is a polynomial of degree 2. Understanding the behavior of rational functions involves analyzing their asymptotes, intercepts, and end behavior.
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Asymptotes

Asymptotes are lines that a graph approaches but never touches. For rational functions, vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe the end behavior of the function. In y = (x⁴ - 1) / x², vertical asymptotes occur at x = 0, and the end behavior can be analyzed by dividing the leading terms.
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Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to numerical long division. It helps simplify rational functions and find oblique asymptotes. For y = (x⁴ - 1) / x², dividing x⁴ by x² gives x², which helps determine the function's behavior as x approaches infinity, indicating the presence of an oblique asymptote.
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Related Practice
Textbook Question

Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


g(x) = √(2x − x²)

Textbook Question

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

74. y' = (x² - 2x)(x - 5)²

Textbook Question

Finding Extreme Values

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.


y = 𝓍³ + 𝓍² ― 8𝓍 + 5

Textbook Question

119. Find the values of constants a, b, and c such that the graph of y = ax^3 + bx^2 + cx has a

local maximum at x = 3, local minimum at x =- 1, and inflection point at (1, 11).

Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


ds/dt = 1 + cos t, s(0) = 4

Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(8y − 2 / y¹ᐟ⁴) dy