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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 12.4.51b
Chapter 12, Problem 12.4.51b

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number.

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Recall that an ellipse can be represented by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are positive constants.
To find the slope of the tangent line at any point on the ellipse, implicitly differentiate the ellipse equation with respect to \(x\): \(\frac{d}{dx} \left( \frac{x^2}{a^2} + \frac{y^2}{b^2} \right) = \frac{d}{dx} (1)\).
Applying the derivative, we get: \(\frac{2x}{a^2} + \frac{2y}{b^2} \frac{dy}{dx} = 0\). Solve for \(\frac{dy}{dx}\) (the slope \(s\)): \(\frac{dy}{dx} = s = -\frac{b^2}{a^2} \frac{x}{y}\).
Notice that the slope depends on both \(x\) and \(y\). For a fixed slope \(s\), rearrange the equation to express \(y\) in terms of \(x\) and \(s\): \(s = -\frac{b^2}{a^2} \frac{x}{y} \implies y = -\frac{b^2}{a^2} \frac{x}{s}\) (assuming \(s \neq 0\)).
Substitute this expression for \(y\) back into the ellipse equation to find the \(x\)-coordinates where the slope is \(s\). This will lead to a quadratic equation in \(x\), which can have zero, one, or two real solutions depending on \(s\). Therefore, the number of points on the ellipse with slope \(s\) can vary, and it is not always exactly two.

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Key Concepts

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

Slope of a Curve and Tangent Lines

The slope of a curve at a point is the derivative of the function defining the curve at that point. It represents the slope of the tangent line to the curve, indicating the instantaneous rate of change. Understanding how to find and interpret slopes is essential for analyzing points where the curve has a specific slope.
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Slopes of Tangent Lines

Parametric and Implicit Differentiation of Ellipses

Ellipses are often defined implicitly or parametrically, requiring implicit differentiation to find the slope of the tangent line. This process involves differentiating both sides of the ellipse equation with respect to x or a parameter, allowing determination of slope values at various points on the ellipse.
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Parameterizing Equations of Circles & Ellipses

Number of Solutions to Slope Equations on Ellipses

Determining how many points on an ellipse have a given slope involves solving an equation derived from the derivative equal to that slope. The number of solutions depends on the ellipse's shape and the slope value, and it is not always exactly two; some slopes may correspond to one, two, or no points, which is key to evaluating the statement.
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Parameterizing Equations of Circles & Ellipses
Related Practice
Textbook Question

Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus.

b. What is the volume of the solid that is generated when R is revolved about the y-axis?

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

Channel flow Water flows in a shallow semicircular channel with inner and outer radii of 1 m and 2 m (see figure). At a point P(r,θ) in the channel, the flow is in the tangential direction (counterclock wise along circles), and it depends only on r, the distance from the center of the semicircles.


a. Express the region formed by the channel as a set in polar coordinates.

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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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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.


a. x = 1 + s, y = 2s and x = 1 + 2t, y = 3t

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Textbook Question

67–72. Derivatives Consider the following parametric curves.

a. Determine dy/dx in terms of t and evaluate it at the given value of t.


x = 2 + 4t, y = 4 − 8t; t = 2

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