교과서 질문
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
f(x) = x(4 − x)³
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
4장, 문제 4.4.27
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
y = 1 / (x² - 1)
검증된 단계별 안내1
Step 1: Analyze the domain of the function y = 1 / (x² - 1). The denominator x² - 1 cannot be zero, so solve x² - 1 = 0 to find the values of x where the function is undefined. These values are x = ±1. The domain of the function is all real numbers except x = ±1.
Step 2: Find the first derivative y' to determine critical points and analyze local extrema. Use the quotient rule for differentiation: if y = u/v, then y' = (u'v - uv') / v². Here, u = 1 and v = x² - 1. Compute y' and simplify.
Step 3: Set the first derivative y' equal to zero to find critical points. Solve the resulting equation for x. Also, check where y' is undefined, as these points may correspond to vertical asymptotes or other critical behavior.
Step 4: Find the second derivative y'' to analyze concavity and locate inflection points. Differentiate y' again using the quotient rule. Set y'' = 0 and solve for x to find potential inflection points. Check the sign of y'' on intervals to determine concavity.
Step 5: Identify absolute extrema by evaluating the function at critical points and endpoints of the domain (if applicable). Also, analyze the behavior of the function as x approaches the vertical asymptotes (x = ±1) and as x approaches ±∞ to understand the overall graph shape.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Graphing Rational Functions
Graphing rational functions involves understanding their behavior based on the numerator and denominator. For the function y = 1 / (x² - 1), we identify vertical asymptotes where the denominator is zero, which occurs at x = ±1. Additionally, we analyze the horizontal asymptote, which is determined by the degrees of the numerator and denominator.
추천 영상:
가이드 코스
5:53
Graph of Sine and Cosine Function
Local Extreme Points
Local extreme points are points on the graph where the function reaches a local maximum or minimum. To find these points, we calculate the first derivative of the function and set it to zero to identify critical points. We then use the first or second derivative test to determine whether these points are maxima, minima, or neither.
추천 영상:
04:50Critical Points
Inflection Points
Inflection points occur where the concavity of the function changes, which can be found by analyzing the second derivative. For the function y = 1 / (x² - 1), we compute the second derivative and set it to zero to find potential inflection points. Evaluating the sign of the second derivative around these points helps confirm whether a change in concavity occurs.
추천 영상:
04:50Critical Points
관련 실천
교과서 질문
110. Suppose the derivative of the function y = f(x) is
y'=(x-1)^22(x-2)(x-4).
At what points, if any, does the graph of f have a local minimum, local maximum, or
point of inflection?
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교과서 질문
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.
f(x) = x − 6√(x − 1)
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교과서 질문
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
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교과서 질문
Finding Functions from Derivatives
In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.
f'(x) = 2x − 1, P(0,0)
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교과서 질문
Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
7. y=sin|x|, -2π≤x≤2π
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