Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.

*x* = 3 ln(*t*),   *y* = 6

*t*

,   *z* = *t*5,   (0, 6, 1)

To find the parametric equations for the tangent line to the given curve at the specified point, we need to follow these steps:

1. **Find the derivatives** of \( x(t) \), \( y(t) \), and \( z(t) \) with respect to \( t \).
2. **Evaluate these derivatives** at the given \( t \)-value corresponding to the point \((0, 6, 1)\).
3. **Use the point and the derivatives** to write the parametric equations of the tangent line.

### Step 1: Derivatives of \( x(t) \), \( y(t) \), and \( z(t) \)

Given:
- \( x(t) = 3 \ln(t) \)
- \( y(t) = 6t \)
- \( z(t) = t^5 \)

Find the derivatives:

1. \( x'(t) = \frac{d}{dt} [3 \ln(t)] = \frac{3}{t} \).
2. \( y'(t) = \frac{d}{dt} [6t] = 6 \).
3. \( z'(t) = \frac{d}{dt} [t^5] = 5t^4 \).

### Step 2: Determine \( t \)-value for the specified point \((0, 6, 1)\)

- For \( x(t) = 0 \):  
  \( 0 = 3 \ln(t) \)  
  \( \ln(t) = 0 \)  
  \( t = e^0 = 1 \).

- For \( y(t) = 6 \):  
  \( 6 = 6t \)  
  \( t = 1 \).

- For \( z(t) = 1 \):  
  \( 1 = t^5 \)  
  \( t = 1 \).

The parameter \( t = 1 \) corresponds to the given point \((0, 6, 1)\).

### Step 3: Evaluate derivatives at \( t = 1 \)

1. \( x'(1) = \frac{3}{1} = 3 \).
2. \( y'(1) = 6 \).
3. \( z'(1) = 5 \times 1^4 = 5 \).

### Step 4: Parametric Equations of the Tangent Line

The parametric equations of the tangent line can be expressed as:

\[
x = x_0 + x'(1) \cdot t = 0 + 3t = 3t
\]

\[
y = y_0 + y'(1) \cdot t = 6 + 6t = 6 + 6t
\]

\[
z = z_0 + z'(1) \cdot t = 1 + 5t = 1 + 5t
\]

So, the parametric equations for the tangent line at the given point are:

\[
\boxed{x = 3t}
\]

\[
\boxed{y = 6 + 6t}
\]

\[
\boxed{z = 1 + 5t}
\]

Do you want further details or have any questions?

**Related Questions:**
1. How do you find the normal line to a parametric curve at a given point?
2. What is the geometric interpretation of parametric equations for tangent lines?
3. How do you find the curvature of a parametric curve at a given point?
4. How do you determine if a parametric curve is smooth at a point?
5. How do you find the arc length of a parametric curve?

**Tip:** For finding tangent lines, always ensure the point corresponds to a specific parameter value.

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point: x = 3 ln(t), y = 6t, z = t^5 at (0, 6, 1).

Explore how to find the parametric equations for the tangent line to the curve x = 3 ln(t), y = 6t, z = t^5 at the point (0, 6, 1). This solution involves calculus concepts such as derivatives and parametric equations. Perfect for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation