Textbook Question
77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.
x = 2 + √t, y = 2 - 4t; slope = -8
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16. Parametric Equations & Polar Coordinates
Calculus with Parametric Curves
8:08 minutesProblem 12.1.84
Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π
Verified step by step guidance1
Recall the formula for the arc length of a parametric curve given by \(x = x(t)\) and \(y = y(t)\) over the interval \(a \leq t \leq b\):
\[L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\]
Find the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) for the given functions:
\[x = e^t \sin t, \quad y = e^t \cos t\]
Use the product rule for differentiation since both \(x\) and \(y\) are products of \(e^t\) and trigonometric functions.
Compute \(\frac{dx}{dt}\):
\[\frac{dx}{dt} = \frac{d}{dt}(e^t \sin t) = e^t \sin t + e^t \cos t\]
Compute \(\frac{dy}{dt}\):
\[\frac{dy}{dt} = \frac{d}{dt}(e^t \cos t) = e^t \cos t - e^t \sin t\]
Substitute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) into the arc length formula under the square root:
\[\sqrt{(e^t \sin t + e^t \cos t)^2 + (e^t \cos t - e^t \sin t)^2}\]
Simplify the expression inside the square root by expanding and combining like terms.
Set up the integral for the arc length over the interval \(0 \leq t \leq 2\pi\):
\[L = \int_0^{2\pi} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt\]
After simplification, evaluate or express the integral in a form ready for calculation.
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Video duration:
8mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations
Parametric equations express the coordinates of a curve as functions of a parameter, often denoted 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 parametric forms is essential for calculating properties like arc length.
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Parameterizing Equations
Arc Length Formula for Parametric Curves
The arc length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is given by the integral of the square root of (dx/dt)² + (dy/dt)² dt. This formula sums the infinitesimal distances along the curve, providing the total length over the interval.
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Differentiation of Exponential and Trigonometric Functions
Calculating dx/dt and dy/dt requires differentiating functions involving products of exponentials and trigonometric terms. Mastery of the product rule and derivatives of e^t, sin t, and cos t is necessary to correctly find these derivatives and proceed with the arc length calculation.
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