Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 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

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.

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.

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

Roots (Zeros)

Show that the functions in Exercises 19–26 have exactly one zero in the given interval.

g(t) = 1/(1 − t) + √(1 + t) − 3.1, (−1, 1)

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Textbook Question

101. In Exercises 101 and 102, the graph of f' is given. Determine x-values corresponding to local minima, local maxima, and inflection points for the graph of f.

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Textbook Question

Root Finding

5. Use Newton's method to find the positive fourth root of 2 by solving the equation x^4 -2 = 0. Start with x_0 = 1 and find x_2.

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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.

71. y' = x(x² - 12)

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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).

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Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

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

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Sketch the graphs of the rational functions in Exercises 53–60.𝓍⁴ ― 1y = ------------------𝓍²

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Key Concepts

Rational Functions

Asymptotes

Polynomial Long Division

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

𝓍⁴ ― 1
y = ------------------
𝓍²