Tangent Lines to Parametrized Curves

In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.

x = t + eᵗ, y = 1 − eᵗ, t = 0

검증된 단계별 안내

First, find the coordinates of the point on the curve at \( t = 0 \) by substituting \( t = 0 \) into the parametric equations \( x = t + e^{t} \) and \( y = 1 - e^{t} \). This gives the point \( (x_0, y_0) \).

Next, compute the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) by differentiating each parametric equation with respect to \( t \).

Then, find the slope of the tangent line \( \frac{dy}{dx} \) at \( t = 0 \) using the chain rule for parametric curves: \( \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \).

Use the point-slope form of a line to write the equation of the tangent line at the point \( (x_0, y_0) \) with slope \( \frac{dy}{dx} \): \[ y - y_0 = \left( \frac{dy}{dx} \right)(x - x_0) \].

To find the second derivative \( \frac{d^2 y}{dx^2} \) at \( t = 0 \), first compute \( \frac{d}{dt} \left( \frac{dy}{dx} \right) \), then divide by \( \frac{dx}{dt} \) using the formula: \[ \frac{d^2 y}{dx^2} = \frac{\frac{d}{dt} \left( \frac{dy}{dx} \right)}{\frac{dx}{dt}} \]. Evaluate this expression at \( t = 0 \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

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주요 개념

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

Parametric Equations and Curves

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves. Understanding how to work with these equations is essential for analyzing the curve's behavior at specific parameter values.

Finding the Tangent Line to a Parametric Curve

The tangent line to a parametric curve at a given t is found by computing the derivatives dx/dt and dy/dt, then using dy/dx = (dy/dt)/(dx/dt) to find the slope. The point on the curve at t gives the coordinates for the tangent point. The tangent line equation uses this slope and point to describe the line touching the curve at that parameter.

Second Derivative d²y/dx² for Parametric Curves

The second derivative d²y/dx² measures the curvature of the parametric curve and is found using the formula d²y/dx² = (d/dt(dy/dx)) / (dx/dt). This requires differentiating the slope dy/dx with respect to t and dividing by dx/dt. It provides insight into the concavity and shape of the curve at the given parameter value.

Tangent Lines to Parametrized CurvesIn Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.x = t + eᵗ, y = 1 − eᵗ, t = 0