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=−t+6, y=3t−3; −5≤t≤5
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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 12.1.11c
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
검증된 단계별 안내1
Start with the given parametric equations: \(x = 2t\) and \(y = 3t - 4\) where \(-10 \leq t \leq 10\).
Solve the first equation for the parameter \(t\): from \(x = 2t\), we get \(t = \frac{x}{2}\).
Substitute \(t = \frac{x}{2}\) into the second equation \(y = 3t - 4\) to eliminate the parameter \(t\). This gives \(y = 3 \left( \frac{x}{2} \right) - 4\).
Simplify the expression to get the relationship between \(x\) and \(y\): \(y = \frac{3}{2}x - 4\).
Describe the curve: since the equation is linear in \(x\) and \(y\), the curve is a straight line with slope \(\frac{3}{2}\) and \(y\)-intercept \(-4\). The parameter range \(-10 \leq t \leq 10\) corresponds to \(x\) values between \(-20\) and \(20\).
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Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted 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
Curve Description from Equations
Describing the curve means interpreting the resulting equation or parametric form to identify its geometric shape, such as a line, circle, or parabola. Understanding the domain of t helps determine the portion of the curve represented.
추천 영상:
04:47Introduction to Parametric Equations
관련 실천
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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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
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.
d. The parametric equations x=cos t, y=sin t, for −π/2≤t≤π/2, describe a semicircle.
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교과서 질문
Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).
c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.
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