Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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**Problem 2:** Find the equation of the tangent line for the following parametric function. Then plot the original curve and the tangent line on the same graph. - \( x = 2 + \sec t \) - \( y = 1 + 2 \tan t \) - \( t = \frac{\pi}{6} \) **Instructions:** 1. **Differentiate the Parametric Equations:** - Compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\). 2. **Find the Slope of the Tangent Line:** - Use the formula for the slope of the tangent line: \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\). 3. **Evaluate at \( t = \frac{\pi}{6} \):** - Substitute \( t = \frac{\pi}{6} \) into the expressions for \( x \) and \( y \) to find the point on the curve. - Substitute \( t = \frac{\pi}{6} \) into the expression for the slope to find the slope at this point. 4. **Find the Equation of the Tangent Line:** - Use the point-slope form of the equation of a line: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the point on the curve. 5. **Graphing:** - Plot the curve using the parametric equations. - Plot the tangent line on the same graph.
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Transcribed Image Text:**Problem 2:** Find the equation of the tangent line for the following parametric function. Then plot the original curve and the tangent line on the same graph. - \( x = 2 + \sec t \) - \( y = 1 + 2 \tan t \) - \( t = \frac{\pi}{6} \) **Instructions:** 1. **Differentiate the Parametric Equations:** - Compute \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\). 2. **Find the Slope of the Tangent Line:** - Use the formula for the slope of the tangent line: \(\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}\). 3. **Evaluate at \( t = \frac{\pi}{6} \):** - Substitute \( t = \frac{\pi}{6} \) into the expressions for \( x \) and \( y \) to find the point on the curve. - Substitute \( t = \frac{\pi}{6} \) into the expression for the slope to find the slope at this point. 4. **Find the Equation of the Tangent Line:** - Use the point-slope form of the equation of a line: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the point on the curve. 5. **Graphing:** - Plot the curve using the parametric equations. - Plot the tangent line on the same graph.
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Step 1: Given the information
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Step 2: Find the point on the curve
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Step 3: Find the slope of the tangent line
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Step 4: Find the equation of tangent line
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Step 5: Graph of the curve
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