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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 12.1.75
Chapter 12, Problem 12.1.75

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=cos t+t sin t,y=sin t−t cos t; t=π/4

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1
Identify the parametric equations given: \(x = \cos t + t \sin t\) and \(y = \sin t - t \cos t\).
Find the derivatives of \(x\) and \(y\) with respect to \(t\): compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) using the product rule where necessary.
Evaluate \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) at \(t = \frac{\pi}{4}\) to find the slope of the tangent line using \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\).
Calculate the coordinates of the point on the curve at \(t = \frac{\pi}{4}\) by substituting \(t\) into the original parametric equations to get \((x_0, y_0)\).
Use the point-slope form of the line equation: \(y - y_0 = m(x - x_0)\), where \(m\) is the slope found in step 3, to write the equation of the tangent line.

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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. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves. Understanding how to work with these equations is essential for finding derivatives and tangent lines.
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Derivative of Parametric Curves

To find the slope of the tangent line to a parametric curve, compute dy/dx by dividing dy/dt by dx/dt, provided dx/dt ≠ 0. This method uses the chain rule and allows determination of the instantaneous rate of change of y with respect to x at a given parameter value.
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Equation of a Tangent Line

Once the slope of the tangent line at a point is found, the tangent line's equation can be written using the point-slope form: y - y₀ = m(x - x₀), where (x₀, y₀) is the point on the curve and m is the slope. This equation represents the line that just touches the curve at that point.
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Related Practice
Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

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Textbook Question

15–22. Sets in polar coordinates Sketch the following sets of points.


r = 3

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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 = 4 cos t, y = 4 sin t; slope = 1/2

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Textbook Question

Multiple descriptions Which of the following parametric equations describe the same curve?

a. x = 2t², y = 4 + t; -4 ≤ t ≤ 4

b. x = 2t⁴, y = 4 + t²; -2 ≤ t ≤ 2

c. x = 2t^(2/3), y = 4 + t^(1/3); -64 ≤ t ≤ 64

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Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = cos t, y = 1 + sin t; 0 ≤ t ≤ 2π

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Textbook Question

69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.

x² = -6y; (-6, -6)

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