10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)

검증된 단계별 안내

Start with the given parametric equations: \(x = \ln t\) and \(y = 8 \ln t^{2}\), where \(1 \leq t \leq e^{2}\).

Recall the logarithm property: \(\ln t^{2} = 2 \ln t\). Use this to rewrite \(y\) as \(y = 8 \times 2 \ln t = 16 \ln t\).

Since \(x = \ln t\), substitute \(\ln t\) in the expression for \(y\) to get \(y = 16x\).

This gives the Cartesian equation relating \(x\) and \(y\) without the parameter \(t\): \(y = 16x\).

Note the domain for \(t\) translates to \(x\) because \(x = \ln t\). Since \(1 \leq t \leq e^{2}\), then \(0 \leq x \leq 2\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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 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.

Eliminating the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove the parameter t, resulting in a direct relationship between x and y. This often requires solving one equation for t and substituting into the other to find an explicit or implicit equation in x and y.

Logarithmic Functions and Their Properties

Logarithmic functions, like ln(t), are the inverses of exponential functions and have properties such as ln(a^b) = b ln(a). Understanding these properties is essential for simplifying expressions and eliminating parameters when the parametric equations involve logarithms.

10–12. Parametric curves a. Eliminate the parameter to obtain an equation in x and y.x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)