Find the parametric equations for the tangent line to the curve æ = t3 – 1,y = to +1, z = t° at the point (0, 2, 1). Use the variable t for your parameter. y =
Find the parametric equations for the tangent line to the curve æ = t3 – 1,y = to +1, z = t° at the point (0, 2, 1). Use the variable t for your parameter. y =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement:**
Find the parametric equations for the tangent line to the curve
\[ x = t^3 - 1, \, y = t^5 + 1, \, z = t^5 \]
at the point \((0, 2, 1)\). Use the variable \( t \) for your parameter.
**Blank Fields for Solutions:**
- \( x = \) [input field],
- \( y = \) [input field],
- \( z = \) [input field],
**Instructions:**
To find the parametric equations of the tangent line, follow these steps:
1. Determine the value of \( t \) at the given point by solving the system:
- \( t^3 - 1 = 0 \),
- \( t^5 + 1 = 2 \),
- \( t^5 = 1 \).
2. Calculate the derivatives \(\frac{dx}{dt}\), \(\frac{dy}{dt}\), and \(\frac{dz}{dt}\) to find the tangent vector at the point.
3. Use the point and the tangent vector to write the parametric equations for the tangent line:
- \( x(t) = x_0 + \frac{dx}{dt} \cdot (t - t_0) \),
- \( y(t) = y_0 + \frac{dy}{dt} \cdot (t - t_0) \),
- \( z(t) = z_0 + \frac{dz}{dt} \cdot (t - t_0) \),
where \((x_0, y_0, z_0)\) is the given point and \((t_0)\) is the parameter corresponding to that point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc247148b-ac88-4d23-ab16-d846cf9acaa7%2F37994b70-a91c-4cc0-84f0-71441f58c802%2Fpafjx4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the parametric equations for the tangent line to the curve
\[ x = t^3 - 1, \, y = t^5 + 1, \, z = t^5 \]
at the point \((0, 2, 1)\). Use the variable \( t \) for your parameter.
**Blank Fields for Solutions:**
- \( x = \) [input field],
- \( y = \) [input field],
- \( z = \) [input field],
**Instructions:**
To find the parametric equations of the tangent line, follow these steps:
1. Determine the value of \( t \) at the given point by solving the system:
- \( t^3 - 1 = 0 \),
- \( t^5 + 1 = 2 \),
- \( t^5 = 1 \).
2. Calculate the derivatives \(\frac{dx}{dt}\), \(\frac{dy}{dt}\), and \(\frac{dz}{dt}\) to find the tangent vector at the point.
3. Use the point and the tangent vector to write the parametric equations for the tangent line:
- \( x(t) = x_0 + \frac{dx}{dt} \cdot (t - t_0) \),
- \( y(t) = y_0 + \frac{dy}{dt} \cdot (t - t_0) \),
- \( z(t) = z_0 + \frac{dz}{dt} \cdot (t - t_0) \),
where \((x_0, y_0, z_0)\) is the given point and \((t_0)\) is the parameter corresponding to that point.
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