교과서 질문
Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.
f(x) = 1/x; Q (-2, 4)
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Ch. 3 - Derivatives
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
3장, 문제 69b
The following equations implicitly define one or more functions.
b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)
검증된 단계별 안내1
Start by rewriting the given equation x + y³ - xy = 1 as y³ - 1 = xy - x. This sets up the equation for factoring.
Notice that the left side y³ - 1 is a difference of cubes, which can be factored using the formula a³ - b³ = (a - b)(a² + ab + b²). Here, a = y and b = 1.
Apply the difference of cubes formula to factor the left side: y³ - 1 = (y - 1)(y² + y + 1).
On the right side, factor out the common factor x from xy - x to get x(y - 1).
Set the factored forms equal: (y - 1)(y² + y + 1) = x(y - 1). To solve for y, consider the cases where y - 1 = 0 and where y - 1 ≠ 0, leading to different solutions for y.
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.
Implicit Functions
Implicit functions are defined by equations where the dependent variable is not isolated on one side. In this case, the equation x + y³ - xy = 1 defines y as a function of x without explicitly solving for y. Understanding how to manipulate these equations is crucial for identifying the relationships between variables.
추천 영상:
가이드 코스
05:14
Finding The Implicit Derivative
Factoring
Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In the context of the given equation, factoring helps to simplify the equation after rewriting it, making it easier to isolate y and identify the functions defined by the equation.
추천 영상:
06:11Limits of Rational Functions: Denominator = 0
Solving for y
Solving for y involves rearranging an equation to express y explicitly in terms of x. This is essential for identifying the functions y = f₁(x), y = f₂(x), etc. In the given problem, after rewriting and factoring the equation, one can isolate y to find the specific forms of the functions defined implicitly by the original equation.
추천 영상:
5:02Solving Logarithmic Equations
관련 실천
교과서 질문
Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = 3x3 + 5x2 + 6x
253
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교과서 질문
Calculate the derivative of the following functions.
y = e^2x(2x-7)⁵
399
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교과서 질문
The following equations implicitly define one or more functions.
a. Find dy/dx using implicit differentiation.
y² = x²(4 − x) / 4 + x (right strophoid)
202
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교과서 질문
The following equations implicitly define one or more functions.
a. Find dy/dx using implicit differentiation.
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)
290
views
교과서 질문
The following equations implicitly define one or more functions.
c. Use the functions found in part (b) to graph the given equation.
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)
271
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