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17Calculus Parametric Equations - Slope & Tangent Lines

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On this page we discuss calculating slope and tangent lines of parametric equations.

To find the equation of a tangent line to a graph given by a set of parametric equations, we need to be able to find the slope by calculating the derivative \( dy/dx \) using the parametric derivative \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \text{ where } dx/dt \neq 0 \] For horizontal tangent lines, the slope \(dy/dx\) is zero, so we need \( dy/dt = 0 \) and \( dx/dt \neq 0 \). For vertical tangent lines, the slope is undefined, which means that \( dx/dt = 0 \) when \( dy/dt \neq 0 \). In the case where both \( dx/dt = 0\) and \( dy/dt = 0 \) at the same point, we need to handle that case separately, since nothing can be concluded from \( dy/dx = 0/0 \), which is indeterminate.

Once you have found the slope, you can easily find the equation of a tangent line. Go to the Tangent Lines page for more information.

Schaum's 3,000 Solved Problems in Calculus

Practice

Unless otherwise instructed, find the equation of the tangent line to these parametric curves at the given points.

\( x=4t \), \( y=3t^2+2 \); \(t=2\).

Problem Statement

Find the equation of the tangent line to the parametric curve \( x=4t \), \( y=3t^2+2 \) at \(t=2\).

Solution

The Organic Chemistry Tutor - 3501 video solution

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\( x=t^2-5 \), \( y=t^2-4t \); \( (-4,-3) \)

Problem Statement

Find the equation of the tangent line to the parametric curve \( x=t^2-5 \), \( y=t^2-4t \) at \( (-4,-3) \).

Solution

The Organic Chemistry Tutor - 3502 video solution

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\( x=8\cos\theta \), \( y=6\sin\theta \); \( \theta = \pi/3 \)

Problem Statement

Find the equation of the tangent line to the parametric curve \( x=8\cos\theta \), \( y=6\sin\theta \) at \( \theta = \pi/3 \).

Solution

The Organic Chemistry Tutor - 3503 video solution

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\(x=2t^2+1\), \(y=3t^3+2\); \(t=1\).

Problem Statement

Find the equation of the tangent line to the parametric curve \(x=2t^2+1\), \(y=3t^3+2\) at \(t=1\).

Solution

Krista King Math - 54 video solution

video by Krista King Math

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\(x=3(t-\sin(t))\), \(y=3(1-\cos(t))\); \(t=\pi/2\).

Problem Statement

Find the equation of the tangent line to the parametric curve \(x=3(t-\sin(t))\), \(y=3(1-\cos(t))\) at \(t=\pi/2\).

Solution

Krista King Math - 55 video solution

video by Krista King Math

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\(x=t\cos(t)\), \(y=t\sin(t)\); \(t=\pi\)

Problem Statement

Find the equation of the tangent line to the parametric curve \(x=t\cos(t)\), \(y=t\sin(t)\) at \(t=\pi\).

Solution

Krista King Math - 467 video solution

video by Krista King Math

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\(x=3t^2-t\), \(y=\sqrt{t}\); \(t=4\).

Problem Statement

Find the equation of the tangent line to the parametric curve \(x=3t^2-t\), \(y=\sqrt{t}\) at \(t=4\).

Solution

PatrickJMT - 1371 video solution

video by PatrickJMT

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Practice Instructions

Unless otherwise instructed, find the equation of the tangent line to these parametric curves at the given points.

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