Implicitly Defined Parametrizations

Assuming that the equations in Exercises 15−20 define x and y implicitly as differentiable functions x=f(t), y=g(t), find the slope of the curve x=f(t), y=g(t) at the given value of t.

x sin t + 2x = t, t sin t − 2t = y, t = π

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Identify the given implicit equations: \(x \sin t + 2x = t\) and \(t \sin t - 2t = y\), where \(x\) and \(y\) are functions of \(t\).

Differentiate both equations with respect to \(t\) using implicit differentiation. For the first equation, apply the product rule to \(x \sin t\) and remember that \(x\) is a function of \(t\): \(\frac{d}{dt}(x \sin t) + \frac{d}{dt}(2x) = \frac{d}{dt}(t)\)

Express the derivatives explicitly: \(\frac{dx}{dt} \sin t + x \cos t + 2 \frac{dx}{dt} = 1\)

Solve this equation for \(\frac{dx}{dt}\), which represents \(x'(t)\), the derivative of \(x\) with respect to \(t\).

For the second equation, differentiate \(y = t \sin t - 2t\) with respect to \(t\) to find \(\frac{dy}{dt}\), then find the slope of the curve at \(t = \pi\) by calculating \(\frac{dy/dt}{dx/dt}\).

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주요 개념

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

Implicit Differentiation

Implicit differentiation is a technique used to find derivatives of functions defined implicitly by equations involving both variables. Instead of solving explicitly for one variable, we differentiate both sides with respect to the independent variable, applying the chain rule to terms involving dependent variables.

Parametric Equations and Derivatives

Parametric equations express coordinates as functions of a parameter, often t. The slope of the curve at a point is found by computing dy/dx = (dy/dt) / (dx/dt), which requires differentiating both x(t) and y(t) with respect to t and then dividing the results.

Chain Rule in Differentiation

The chain rule allows differentiation of composite functions by multiplying the derivative of the outer function by the derivative of the inner function. It is essential when differentiating implicit functions or parametric forms where variables depend on a parameter.

Implicitly Defined ParametrizationsAssuming that the equations in Exercises 15−20 define x and y implicitly as differentiable functions x=f(t), y=g(t), find the slope of the curve x=f(t), y=g(t) at the given value of t.x sin t + 2x = t, t sin t − 2t = y, t = π