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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh.12 - Parametric and Polar Curves문제 12.1.10
12장, 문제 12.1.10

Use calculus to find the arc length of the line segment x=3t+1, y=4t, for 0≤t≤1. Check your work by finding the distance between the endpoints of the line segment. 

검증된 단계별 안내
1
Identify the parametric equations given: \(x(t) = 3t + 1\) and \(y(t) = 4t\), with the parameter \(t\) ranging from 0 to 1.
Find the derivatives of \(x(t)\) and \(y(t)\) with respect to \(t\): compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\).
Use the arc length formula for parametric curves: \(L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\), where \(a=0\) and \(b=1\).
Substitute the derivatives into the integral and simplify the expression inside the square root to prepare for integration.
Evaluate the integral to find the arc length, then verify your result by calculating the straight-line distance between the endpoints \((x(0), y(0))\) and \((x(1), y(1))\) using the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. In this problem, x and y are given in terms of t, allowing us to analyze the curve's properties by differentiating with respect to t.
추천 영상:
08:02
Parameterizing Equations

Arc Length Formula for Parametric Curves

The arc length of a curve defined parametrically by x(t) and y(t) from t=a to t=b is found by integrating the square root of the sum of the squares of the derivatives: ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt. This formula calculates the length of the curve between two parameter values.
추천 영상:
가이드 코스
06:29
Arc Length of Parametric Curves

Distance Between Two Points

The distance between two points (x₁, y₁) and (x₂, y₂) in the plane is given by the formula √((x₂ - x₁)² + (y₂ - y₁)²). This provides a way to verify the arc length for a straight line segment by comparing it to the direct distance between endpoints.
추천 영상:
02:59
Finding Area Between Curves that Cross on the Interval
관련 실천
교과서 질문

Air drop A plane traveling horizontally at 80 m/s over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given by

x = 80t, y = −4.9t² + 3000, t ≥ 0

where the origin is the point on the ground directly beneath the plane at the moment of the release (see figure). Graph the trajectory of the packet and find the coordinates of the point where the packet lands.

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교과서 질문

41–44. Intersection points and area  Find all the intersection points of the following curves. Find the area of the entire region that lies within both curves


r = 3 sin θ and r = 3 cos θ

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교과서 질문

102–104. Spirals Graph the following spirals. Indicate the direction in which the spiral is generated as θ increases, where θ>0. Let a=1 and a=−1.


Spiral of Archimedes: r = aθ

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교과서 질문

80–83. Equations of circles Use the results of Exercises 78–79 to describe and graph the following circles.


r² - 8r cos(θ - π/2) = 9

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교과서 질문

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r = 3 csc θ

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교과서 질문

49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.


y = x²

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