교과서 질문
Use the guidelines of this section to make a complete graph of f.
f(x) = tan⁻¹ (x²/√3)
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
4장, 문제 31
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x²/(x - 2)
검증된 단계별 안내1
Identify the domain of the function f(x) = x²/(x - 2). The function is undefined where the denominator is zero, so set x - 2 = 0 and solve for x to find the vertical asymptote at x = 2.
Determine the x-intercepts by setting the numerator equal to zero: x² = 0. Solve for x to find the x-intercept at x = 0.
Find the y-intercept by evaluating the function at x = 0: f(0) = 0²/(0 - 2). This confirms the y-intercept is also at (0, 0).
Analyze the end behavior of the function by considering the limits as x approaches positive and negative infinity. Since the degree of the numerator is greater than the degree of the denominator, the function will have an oblique asymptote. Perform polynomial long division to find the equation of the oblique asymptote.
Sketch the graph using the information gathered: plot the intercepts, draw the vertical asymptote at x = 2, and sketch the behavior near the asymptotes and at infinity, ensuring the graph approaches the oblique asymptote as x goes to positive or negative infinity.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
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Function Behavior
Understanding the behavior of the function f(x) = x²/(x - 2) is crucial. This includes identifying its domain, range, and any asymptotes. The function is undefined at x = 2, which creates a vertical asymptote, and analyzing the limits as x approaches this value helps in understanding the graph's behavior near the asymptote.
추천 영상:
5:46
Graphs of Exponential Functions
Critical Points and Extrema
Finding critical points involves taking the derivative of the function and setting it to zero. This helps identify local maxima and minima, which are essential for sketching the graph accurately. Analyzing the first derivative test can also indicate where the function is increasing or decreasing.
추천 영상:
04:50Critical Points
End Behavior
End behavior describes how the function behaves as x approaches positive or negative infinity. For rational functions like f(x) = x²/(x - 2), examining the leading terms helps predict the horizontal asymptote and overall shape of the graph. This understanding is vital for completing the graph accurately.
추천 영상:
03:07Cases Where Limits Do Not Exist
관련 실천
교과서 질문
Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = (x⁴/2) - 3x² + 4x + 1
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교과서 질문
Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = 3x/(x² + 3)
174
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교과서 질문
Demand and elasticity The economic advisor of a large tire store proposes the demand function D(p) = 1800/p-40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p.
c. Find the elasticity function on the domain of the demand function.
293
views
교과서 질문
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = 3x/(x² - 1)
229
views
교과서 질문
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = (x2 + 12)/(2x + 1)
269
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