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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 12.1.67b
Chapter 12, Problem 12.1.67b

67–72. Derivatives Consider the following parametric curves.
b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of t.


x = 2 + 4t, y = 4 − 8t; t = 2

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Identify the parametric equations given: \(x = 2 + 4t\) and \(y = 4 - 8t\).
Find the coordinates of the point on the curve at \(t = 2\) by substituting \(t = 2\) into both equations: calculate \(x(2)\) and \(y(2)\).
Compute the derivatives of \(x\) and \(y\) with respect to \(t\): find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\).
Determine the slope of the tangent line at \(t = 2\) using the formula for parametric curves: \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\) evaluated at \(t = 2\).
Write the equation of the tangent line in point-slope form using the point \((x(2), y(2))\) and the slope found in the previous step: \(y - y(2) = m (x - x(2))\).

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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, usually t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves and motions.
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Derivative of Parametric Curves

To find the slope of the tangent line to a parametric curve, compute dx/dt and dy/dt, then find dy/dx = (dy/dt) / (dx/dt). This derivative gives the instantaneous rate of change of y with respect to x at a given parameter value.
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Equation of the Tangent Line

Once the slope of the tangent line at a point is known, use the point-slope form y - y₀ = m(x - x₀) to write the tangent line's equation. Here, (x₀, y₀) is the point on the curve at the given parameter t, and m is the slope.
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Related Practice
Textbook Question

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).

b. Explain why tan θ = y/x.


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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. The parametric equations x=t, y=t², for t≥0, describe the complete parabola y=x².

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Textbook Question

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.


b. x = 2 + 5s, y = 1 + s and x = 4 + 10t, y = 3 + 2t

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Textbook Question

Spiral arc length Consider the spiral r=4θ, for θ≥0.


c. Show that L′(θ)>0. Is L″(θ) positive or negative? Interpret your answer.

Textbook Question

Regions bounded by a spiral: Let Rₙ be the region bounded by the nth turn and the (n+1)st turn of the spiral r = e⁻ᶿ in the first and second quadrants, for θ ≥ 0 (see figure).

c. Evaluate lim(n→∞) Aₙ₊₁/Aₙ.


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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. An object following the parametric curve x=2cos 2πt, y=2 sin 2πt circles the origin once every 1 time unit.

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