교과서 질문
Find the derivatives of the functions in Exercises 1–42.
__
𝔂 = ( √ x )²
( 1 + x )
273
views
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
당신이 사용하는 게 아니라요?교과서 변경다른 교과서를 선택하세요
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
3장, 문제 3.7.49
Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?
검증된 단계별 안내1
Step 1: Begin by understanding the geometry of the problem. The parabola is given as x = y², and the point (a, 0) lies on the x-axis. A normal line to the parabola is perpendicular to the tangent line at a given point on the curve. The goal is to show that three normal lines can be drawn from (a, 0) to the parabola only if a > 1/2, and to find the value of a for which the other two normal lines are perpendicular.
Step 2: Find the slope of the tangent line to the parabola at any point (x, y). The derivative of x = y² with respect to y is dx/dy = 2y. The slope of the tangent line is therefore 2y. The slope of the normal line, being perpendicular to the tangent line, is -1/(2y).
Step 3: Write the equation of the normal line passing through a point (x₀, y₀) on the parabola. The equation is given by: x - x₀ = (-1/(2y₀))(y - y₀). Substitute x₀ = y₀² (from the parabola equation) into this normal line equation.
Step 4: To find the intersection of the normal line with the point (a, 0), substitute x = a and y = 0 into the normal line equation derived in Step 3. This will yield a relationship between a and y₀. Solve this equation to determine the values of y₀ for which the normal line passes through (a, 0).
Step 5: Analyze the resulting equation to determine the conditions under which three solutions for y₀ exist. This corresponds to three normal lines passing through (a, 0). Show that this is possible only if a > 1/2. Then, determine the specific value of a for which the other two normal lines are perpendicular by examining the slopes of these lines and ensuring their product equals -1.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
10m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Normal Line to a Curve
A normal line to a curve at a given point is a line perpendicular to the tangent line at that point. For a curve defined by a function, the slope of the normal line is the negative reciprocal of the slope of the tangent line. In the context of the parabola x = y², finding the normal line involves determining the derivative to get the tangent slope and then using it to find the normal slope.
추천 영상:
05:13
Slopes of Tangent Lines
Parabola and its Properties
A parabola is a symmetric curve defined by a quadratic equation. In this case, the parabola is given by x = y², which opens to the right. Key properties include its vertex, axis of symmetry, and the fact that it is symmetric about the x-axis. Understanding these properties helps in analyzing how lines, such as normals, interact with the parabola.
추천 영상:
7:42Properties of Parabolas
Perpendicular Lines
Two lines are perpendicular if the product of their slopes is -1. This concept is crucial when determining conditions under which two normal lines to a curve are perpendicular. In the problem, it involves setting up equations based on the slopes of the normal lines and solving for the parameter a to find when the lines are perpendicular.
추천 영상:
05:13Slopes of Tangent Lines
관련 실천
교과서 질문
In Exercises 41–58, find dy/dt.
y = (t⁻³/⁴ sin(t))⁴/³
250
views
교과서 질문
Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).
281
views
교과서 질문
In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.
(y - x)² = 2x + 4, (6, 2)
302
views
교과서 질문
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the surface area S = 6x² of a cube when the edge lengths change from x₀ to x₀ + dx
204
views
교과서 질문
Implicit Differentiation
In Exercises 43–50, find by implicit differentiation.
xy + 2x + 3y = 1
250
views