Textbook Question
Centroids
Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0 ≤ t ≤ π.
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Ch. 11 - Parametric Equations and Polar Coordinates
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 11, Problem 11.2.6
Tangent Lines to Parametrized Curves
In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.
x = sec² t − 1, y = tan t, t = −π/4
Verified step by step guidance1
First, find the derivatives of the parametric equations with respect to the parameter \(t\). Compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) from the given functions \(x = \sec^{2} t - 1\) and \(y = \tan t\).
Next, find the slope of the tangent line \(\frac{dy}{dx}\) at the given value of \(t = -\frac{\pi}{4}\) by using the chain rule for parametric curves: \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\).
Evaluate \(x\) and \(y\) at \(t = -\frac{\pi}{4}\) to find the specific point on the curve where the tangent line touches.
Use the point-slope form of a line, \(y - y_0 = m(x - x_0)\), where \(m\) is the slope found in step 2 and \((x_0, y_0)\) is the point found in step 3, to write the equation of the tangent line.
To find \(\frac{d^{2}y}{dx^{2}}\) at \(t = -\frac{\pi}{4}\), first compute \(\frac{d}{dt}\left(\frac{dy}{dx}\right)\), then divide by \(\frac{dx}{dt}\) using the formula \(\frac{d^{2}y}{dx^{2}} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}\), and finally evaluate at \(t = -\frac{\pi}{4}\).
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11mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations and Curves
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Understanding how x and y depend on t allows us to analyze the curve's shape and behavior by varying t, rather than expressing y directly as a function of x.
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Parameterizing Equations
Finding the Tangent Line to a Parametric Curve
The tangent line at a point on a parametric curve is found using derivatives dx/dt and dy/dt. The slope of the tangent line is dy/dx = (dy/dt) / (dx/dt), evaluated at the given t. Using this slope and the point coordinates, the tangent line equation can be written in point-slope form.
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06:49Differentiation of Parametric Curves
Second Derivative d²y/dx² for Parametric Curves
The second derivative d²y/dx² measures the curvature of the parametric curve and is found by differentiating dy/dx with respect to x. Using the chain rule, d²y/dx² = (d/dt(dy/dx)) / (dx/dt). Evaluating this at the given t provides the curve's concavity at that point.
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06:40Higher Order Derivatives of Parametric Curves
Related Practice
Textbook Question
Finding Cartesian from Parametric Equations
In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.
x = cos t, y = sin 3t
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Textbook Question
Finding Lengths of Polar Curves
Find the lengths of the curves in Exercises 21–28.
The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4
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Textbook Question
Shifting Conic Sections
You may wish to review Section 1.2 before solving Exercises 39-56.
Exercises 53-56 give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes.
x²/4 − y²/5 = 1, right 2, up 2
Textbook Question
Hyperbolas
Exercises 27-34 give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch.
8x² − 2y² = 16
Textbook Question
Polar to Cartesian Equations
Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.
r = 3 cos θ
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