교과서 질문
Graph the functions in Exercises 37–56.
y = (x + 1)²/³
207
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Ch. 1 - Functions
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
1장, 문제 1.1.14
Finding Formulas for Functions
Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.
검증된 단계별 안내1
Start by identifying the given point (x, y) on the graph of the function y = √(x − 3). This implies that x can be expressed in terms of y as x = y^2 + 3.
Recognize that the distance L between the points (x, y) and (4, 0) can be calculated using the distance formula: L = √((x - 4)^2 + (y - 0)^2).
Substitute the expression for x from step 1 into the distance formula: L = √(((y^2 + 3) - 4)^2 + y^2).
Simplify the expression inside the square root: ((y^2 + 3) - 4) simplifies to (y^2 - 1). Therefore, the expression becomes L = √((y^2 - 1)^2 + y^2).
Further simplify the expression: Expand (y^2 - 1)^2 to get y^4 - 2y^2 + 1. Thus, L = √(y^4 - 2y^2 + 1 + y^2), which simplifies to L = √(y^4 - y^2 + 1).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
4m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Distance Formula
The distance formula calculates the distance between two points in a Cartesian plane. For points (x1, y1) and (x2, y2), the distance L is given by L = √((x2 - x1)² + (y2 - y1)²). In this problem, we will use this formula to find the distance between the point (x, y) on the curve and the point (4, 0).
추천 영상:
가이드 코스
5:04
Converting between Degrees & Radians
Function Representation
A function represents a relationship between two variables, typically denoted as y = f(x). In this context, we need to express the distance L as a function of y, which may involve substituting y into the distance formula and rearranging the equation to isolate L. This process is essential for understanding how L varies with y.
추천 영상:
06:21Properties of Functions
Substitution in Functions
Substitution is a technique used in algebra and calculus to replace a variable with another expression. In this case, since y = √(x - 3), we can express x in terms of y by rearranging the equation to x = y² + 3. This substitution allows us to rewrite the distance L solely in terms of y, facilitating the analysis of the relationship between L and y.
추천 영상:
04:27Substitution With an Extra Variable
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