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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.1.73
Chapter 12, Problem 12.1.73

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.


x=t ²−1, y=t ³ +t; t=2

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Identify the parametric equations given: \(x = t^{2} - 1\) and \(y = t^{3} + t\), and the value of the parameter \(t = 2\) at which we want the tangent line.
Find the derivatives of \(x\) and \(y\) with respect to \(t\): compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\). For \(x = t^{2} - 1\), \(\frac{dx}{dt} = 2t\). For \(y = t^{3} + t\), \(\frac{dy}{dt} = 3t^{2} + 1\).
Calculate the slope of the tangent line \(\frac{dy}{dx}\) at \(t=2\) using the chain rule for parametric equations: \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\) evaluated at \(t=2\).
Find the coordinates of the point on the curve at \(t=2\) by substituting \(t=2\) into the parametric equations: \(x(2)\) and \(y(2)\).
Use the point-slope form of the line equation with the point \((x(2), y(2))\) and slope \(\frac{dy}{dx}\) to write the equation of the tangent line: \(y - y(2) = m (x - x(2))\), where \(m\) is the slope found in the previous step.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Here, x and y are given in terms of t, allowing us to analyze the curve's behavior by varying t.
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Derivative of Parametric Curves

To find the slope of the tangent line to a parametric curve, we compute dy/dx as (dy/dt) divided by (dx/dt). This ratio gives the instantaneous rate of change of y with respect to x at a specific parameter value.
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Equation of a Tangent Line

Once the slope of the tangent line and the point of tangency are known, the tangent line's equation can be written using the point-slope form: y - y₀ = m(x - x₀), where m is the slope and (x₀, y₀) is the point on the curve.
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Related Practice
Textbook Question

37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.


The lower half of the circle centered at (−2, 2) with radius 6, oriented in the counterclockwise direction

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Textbook Question

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.


x = 2 cos t, y = 8 sin t; slope = -1

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Textbook Question

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=t,y= √(4−t²) a

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Textbook Question

33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.


The region inside the circle r = 8 sin θ

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Textbook Question

11–20. Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points.


r = 2θ; (π/2, π/4)

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Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.


r = 2 - 2 sin θ  b