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.
∫(x + 1) dx
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.60
Sketch the graphs of the rational functions in Exercises 53–60.
𝓍²
y = ------------------
𝓍² ― 4
Verified step by step guidance1
Identify the rational function: \( f(x) = \frac{x^2}{x^2 - 4} \). This function is a ratio of two polynomials.
Determine the domain of the function. The denominator \( x^2 - 4 \) cannot be zero, so solve \( x^2 - 4 = 0 \) to find the values of \( x \) that are not in the domain.
Find the vertical asymptotes by setting the denominator equal to zero: \( x^2 - 4 = 0 \). Solve for \( x \) to find the vertical asymptotes at \( x = 2 \) and \( x = -2 \).
Determine the horizontal asymptote by comparing the degrees of the numerator and the denominator. Since both are degree 2, the horizontal asymptote is \( y = \frac{1}{1} = 1 \).
Analyze the behavior of the function around the asymptotes and intercepts. Check the sign of the function in the intervals determined by the vertical asymptotes and plot key points to sketch the graph.
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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, where the numerator and the denominator are polynomials. In the given function y = x² / (x² - 4), x² is the numerator and x² - 4 is the denominator. Understanding the behavior of rational functions involves analyzing their asymptotes, intercepts, and domain.
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Intro to Rational Functions
Vertical Asymptotes
Vertical asymptotes occur in rational functions where the denominator equals zero, causing the function to approach infinity. For y = x² / (x² - 4), setting the denominator x² - 4 = 0 gives x = ±2. These values are where the vertical asymptotes occur, indicating the function's undefined points and guiding the graph's behavior near these lines.
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Horizontal Asymptotes
Horizontal asymptotes describe the behavior of a function as x approaches infinity. For rational functions, if the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of their leading coefficients. In y = x² / (x² - 4), both the numerator and denominator have the same degree, so the horizontal asymptote is y = 1, indicating the function's end behavior.
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Related Practice
Textbook Question
Finding Extrema from Graphs
In Exercises 11–14, match the table with a graph.
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.
∫7sin(θ/3) dθ
Textbook Question
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = {x² − x, −2 ≤ x ≤−1
2x² − 3x − 3, −1 < x ≤ 0
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Textbook Question
56. Airplane landing path An airplane is flying at altitude H when it begins its descent to an airport runway that is at horizontal ground distance L from the airplane, as shown in the accompanying figure. Assume that the landing path of the airplane is the graph of a cubic polynomial function y = ax^3+bx^2+cx+d, where y(-L)= H and y(0)=0.
a. What is dy/dx at x = 0?
b. What is dy/dx at x = -L?
c. Use the values for dy/dx at x = 0 and x =- L together with y(0) = 0 and y(-L) = H to show that y(x)=H[2(x/L)^3+3(x/L)^2]
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Textbook Question
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
g(x) = x√8 − x²