 Find the unit vectors that are parallel to the tangent line to the parabola y = x^2 at the point (5, 25).
smaller i-component
larger i-component 

To find the unit vectors parallel to the tangent line of the parabola \( y = x^2 \) at the point \( (5, 25) \), follow these steps:

### Step 1: Find the slope of the tangent line
The slope of the tangent line to the parabola at a given point is determined by the derivative of the function \( y = x^2 \).

\[
\frac{dy}{dx} = 2x
\]

At the point \( x = 5 \):

\[
\frac{dy}{dx} = 2(5) = 10
\]

So, the slope of the tangent line at the point \( (5, 25) \) is 10.

### Step 2: Write the direction vector of the tangent line
The slope of 10 means that the direction vector of the tangent line can be expressed as \( \vec{v} = \langle 1, 10 \rangle \), because the rise-over-run ratio (slope) is 10.

### Step 3: Find the unit vector
To find the unit vectors parallel to the tangent line, we need to normalize the direction vector \( \langle 1, 10 \rangle \). The magnitude of the vector \( \langle 1, 10 \rangle \) is:

\[
|\vec{v}| = \sqrt{1^2 + 10^2} = \sqrt{1 + 100} = \sqrt{101}
\]

The unit vector in the same direction as \( \langle 1, 10 \rangle \) is:

\[
\vec{u_1} = \frac{1}{\sqrt{101}} \langle 1, 10 \rangle = \left\langle \frac{1}{\sqrt{101}}, \frac{10}{\sqrt{101}} \right\rangle
\]

### Step 4: Find the unit vector in the opposite direction
The unit vector in the opposite direction is:

\[
\vec{u_2} = \frac{-1}{\sqrt{101}} \langle 1, 10 \rangle = \left\langle \frac{-1}{\sqrt{101}}, \frac{-10}{\sqrt{101}} \right\rangle
\]

### Step 5: Identify the smaller and larger i-components
- The smaller i-component comes from \( \vec{u_2} = \left\langle \frac{-1}{\sqrt{101}}, \frac{-10}{\sqrt{101}} \right\rangle \).
- The larger i-component comes from \( \vec{u_1} = \left\langle \frac{1}{\sqrt{101}}, \frac{10}{\sqrt{101}} \right\rangle \).

Thus:
- The **smaller i-component** unit vector is \( \left\langle \frac{-1}{\sqrt{101}}, \frac{-10}{\sqrt{101}} \right\rangle \).
- The **larger i-component** unit vector is \( \left\langle \frac{1}{\sqrt{101}}, \frac{10}{\sqrt{101}} \right\rangle \).

Would you like further details or have any questions?

### Related Questions:
1. What is the significance of unit vectors in vector calculus?
2. How do you calculate the magnitude of a vector in 3D space?
3. Can the derivative of a function always be used to find the slope of a tangent line?
4. How would you find the normal vector to the curve \( y = x^2 \) at \( (5, 25) \)?
5. What are the applications of tangent lines in physics?

**Tip**: Unit vectors are often used to indicate direction while ignoring magnitude, which is useful in physics and engineering.

Find the unit vectors that are parallel to the tangent line to the parabola y = x^2 at the point (5, 25).

Discover how to find unit vectors that are parallel to the tangent line of the parabola y = x^2 at the point (5, 25). Learn about differentiation, vectors, and how to calculate unit vectors. This problem is ideal for high school students in Grades 10-12.

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