https://file.aiquickdraw.com/mathbot/file/1729553488348llfhoiqw.jpg

The problem asks to find the equation of the tangent line to the graph of the function \( f(x) = \frac{2x^3 - 1}{x^2} \) at the point \( (1,1) \).

### Steps to solve:

1. **Simplify the function**:
   \[
   f(x) = \frac{2x^3 - 1}{x^2} = 2x - \frac{1}{x^2}.
   \]
   
2. **Find the derivative \( f'(x) \)** to determine the slope of the tangent line:
   We use the sum and power rules to differentiate:
   \[
   f'(x) = \frac{d}{dx}(2x) - \frac{d}{dx}\left(\frac{1}{x^2}\right).
   \]
   - The derivative of \( 2x \) is \( 2 \).
   - The derivative of \( \frac{1}{x^2} \) is \( -2x^{-3} \), which simplifies to \( \frac{2}{x^3} \).

   Therefore,
   \[
   f'(x) = 2 + \frac{2}{x^3}.
   \]

3. **Evaluate \( f'(x) \) at \( x = 1 \)** to get the slope of the tangent line at the point \( (1,1) \):
   \[
   f'(1) = 2 + \frac{2}{1^3} = 2 + 2 = 4.
   \]
   So, the slope of the tangent line is \( 4 \).

4. **Use the point-slope form** of the equation of a line to find the tangent line:
   The point-slope form is:
   \[
   y - y_1 = m(x - x_1),
   \]
   where \( m \) is the slope and \( (x_1, y_1) \) is the point. Here, \( m = 4 \) and \( (x_1, y_1) = (1, 1) \).

   Substituting into the point-slope form:
   \[
   y - 1 = 4(x - 1).
   \]

5. **Simplify** the equation:
   \[
   y - 1 = 4x - 4,
   \]
   \[
   y = 4x - 3.
   \]

### Final Answer:
The equation of the tangent line to the function at \( (1, 1) \) is:
\[
y = 4x - 3.
\]

---

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. How do you find the derivative of more complicated rational functions?
2. What is the geometric meaning of a tangent line to a curve?
3. How do you find the normal line to a curve at a given point?
4. Can you find the equation of the tangent line to a parabola at any point?
5. How would the process differ if the function was implicitly defined?

**Tip:** When finding tangent lines, always evaluate the derivative at the given point to get the slope before using the point-slope form of the equation of a line.

Find the equation of the tangent line to the graph of the function f at (1,1) if f(x) = (2x^3 - 1) / x^2.

Learn how to find the equation of the tangent line to the function f(x) = (2x^3 - 1) / x^2 at the point (1,1). This solution involves calculus concepts such as derivatives and the point-slope form of a line. Suitable for high school students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation