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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 12.3.22
Chapter 12, Problem 12.3.22

Tangent line at the origin Find the polar equation of the line tangent to the polar curve r=4cosθ at the origin. Explain why the slope of this line is undefined.

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1
Recall that the polar curve is given by \(r = 4 \cos \theta\). The point at the origin corresponds to \(r = 0\). Find the values of \(\theta\) where \(r = 0\) by solving \(4 \cos \theta = 0\).
Identify the points on the curve where it passes through the origin. These occur at the values of \(\theta\) found in the previous step. For each such \(\theta\), the curve passes through the origin \((r=0)\).
To find the slope of the tangent line in Cartesian coordinates, convert the polar coordinates to Cartesian using \(x = r \cos \theta\) and \(y = r \sin \theta\). Then express \(y\) as a function of \(x\) implicitly or find \(\frac{dy}{dx}\) using the chain rule and derivatives with respect to \(\theta\).
Use the formula for the slope of the tangent line to a polar curve: \(\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}\). Compute \(\frac{dr}{d\theta}\) and substitute \(r\), \(\theta\), and \(\frac{dr}{d\theta}\) at the point where \(r=0\).
Evaluate the slope at the origin. If the denominator of \(\frac{dy}{dx}\) is zero while the numerator is nonzero, the slope is undefined, indicating a vertical tangent line. This explains why the slope of the tangent line at the origin is undefined.

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Key Concepts

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

Polar Coordinates and Polar Curves

Polar coordinates represent points using a radius and an angle (r, θ) from the origin. Polar curves are equations relating r and θ, describing shapes in the plane. Understanding how to interpret and plot these curves is essential for analyzing their properties, such as tangents.
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Intro to Polar Coordinates

Tangent Lines in Polar Coordinates

The tangent line to a polar curve at a point can be found by converting the curve to Cartesian coordinates or by using the derivative dr/dθ. The slope of the tangent line is given by dy/dx, which can be expressed in terms of r, θ, and dr/dθ. This helps identify the direction and slope of the tangent line at specific points.
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Intro to Polar Coordinates

Undefined Slope and Vertical Tangents

A slope is undefined when the tangent line is vertical, meaning the change in x is zero while y changes. In polar curves, this occurs when the derivative dy/dx approaches infinity or is not defined. Recognizing when the slope is undefined explains the nature of the tangent line, especially at points like the origin.
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Slopes of Tangent Lines
Related Practice
Textbook Question

53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.


A bicyclist rides counterclockwise with constant speed around a circular velodrome track with a radius of 50 m, completing one lap in 24 seconds.

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Textbook Question

Express the polar equation r=f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.

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. 

Textbook Question

What is the polar equation of the horizontal line y = 5?

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 = r − 1, y = r³; −4 ≤ r ≤ 4

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 = 8 + 2t, y = 1; −∞ < t < ∞