교과서 질문
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. The parametric equations x=t, y=t², for t≥0, describe the complete parabola y=x².
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Ch.12 - Parametric and Polar Curves
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
12장, 문제 12.1.13c
11–14. Working with parametric equations Consider the following parametric equations.
c. Eliminate the parameter to obtain an equation in x and y.
d. Describe the curve.
x=−t+6, y=3t−3; −5≤t≤5
검증된 단계별 안내1
Start with the given parametric equations: \(x = -t + 6\) and \(y = 3t - 3\) with the parameter \(t\) in the interval \([-5, 5]\).
Solve the first equation for \(t\): from \(x = -t + 6\), rearrange to get \(t = 6 - x\).
Substitute this expression for \(t\) into the second equation: \(y = 3(6 - x) - 3\).
Simplify the equation for \(y\) to express it solely in terms of \(x\): \(y = 18 - 3x - 3\), which simplifies further to \(y = 15 - 3x\).
Interpret the resulting equation \(y = 15 - 3x\): this is a linear equation representing a straight line. The parameter interval \(-5 \leq t \leq 5\) will restrict the portion of the line that corresponds to the curve.
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.
Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.
추천 영상:
08:02
Parameterizing Equations
Eliminating the Parameter
Eliminating the parameter involves manipulating the parametric equations to remove t, resulting in a direct relationship between x and y. This often requires solving one equation for t and substituting into the other, yielding a Cartesian equation of the curve.
추천 영상:
05:59Eliminating the Parameter
Analyzing and Describing the Curve
Describing the curve means interpreting the resulting Cartesian equation or parametric form to identify its shape, type (line, circle, etc.), and domain restrictions. Understanding the parameter range helps determine the portion of the curve represented.
추천 영상:
11:41Summary of Curve Sketching
관련 실천
교과서 질문
Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.
b. x = 2 + 5s, y = 1 + s and x = 4 + 10t, y = 3 + 2t
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교과서 질문
Spiral arc length Consider the spiral r=4θ, for θ≥0.
c. Show that L′(θ)>0. Is L″(θ) positive or negative? Interpret your answer.
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교과서 질문
11–14. Working with parametric equations Consider the following parametric equations.
c. Eliminate the parameter to obtain an equation in x and y.
d. Describe the curve.
x=2 t,y=3t−4;−10≤t≤10
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교과서 질문
Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.
c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t
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교과서 질문
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number.
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