Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π
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16. Parametric Equations & Polar Coordinates
Calculus with Parametric Curves
11:15 minutesProblem 12.1.75
Textbook Question
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
Verified step by step guidance1
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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11mKey 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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Parameterizing Equations
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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