Ch. 11 - Parametric Equations and Polar Coordinates

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 11 - Parametric Equations and Polar Coordinates

Problema 11.2.18

Capitolo 11, Problema 11.2.18

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

Guida verificata passo dopo passo

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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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.

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Percorso guidato

05:14

Finding The Implicit Derivative

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.

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08:02

Parameterizing Equations

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.

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Intro to the Chain Rule

Pratica correlata

Domanda del libro di testo

Finding Cartesian from Parametric Equations

Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π

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Domanda del libro di testo

Symmetries and Polar Graphs

Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.

r = 1 + 2 sin θ

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Domanda del libro di testo

Finding Polar Areas

Find the areas of the regions in Exercises 9–18.

Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ

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