https://r2.erweima.ai/mathbot/image/17267414865507eu8ot9b.jpg

The problem is written in Japanese and involves the function \( f(x) = x^2 - 2x + 2 \). Let’s go through each part step by step:

1. **Given:**
   - The function \( f(x) = x^2 - 2x + 2 \).
   - A curve \( C: y = f(x) \).

### Let's break down the questions:

1. **(1) Find the equation of the tangent line to the curve at a point \( (a, f(a)) \).**
   - To find the tangent line at the point \( (a, f(a)) \), we need the derivative of \( f(x) \).

   - **Derivative:**  
     \( f'(x) = 2x - 2 \).  
     The slope of the tangent line at \( x = a \) is \( f'(a) = 2a - 2 \).

   - **Equation of the tangent line:**  
     Using the point-slope form of the line, the equation of the tangent line at \( (a, f(a)) \) is:  
     \[ y - f(a) = (2a - 2)(x - a). \]

2. **(2) Find all possible values of \( a \) such that the tangent line passes through the point \( P(0, -2) \).**
   - The tangent line passes through \( (0, -2) \). Plug this point into the tangent line equation:
     \[ -2 - f(a) = (2a - 2)(0 - a). \]
   - **Simplify this equation to find \( a \).**

3. **(3) Find the area enclosed by the curve \( C \), the tangent lines found in (2), and other relevant boundaries.**
   - This involves using definite integrals and the equations of the lines found in part (2) to determine the enclosed area.

I will start by solving part (1) and part (2) step-by-step. Let's go ahead and find the tangent line equation and the specific values for \( a \).It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can still help guide you through the steps manually or provide explanations for each part of the problem. 

Here are a few things you could try in the meantime:
1. Calculate the derivative \( f'(x) \) to find the slope of the tangent line.
2. Use the point-slope formula to write the equation of the tangent line at a point \( (a, f(a)) \).
3. Set up the equation for the tangent line passing through \( (0, -2) \) and solve for \( a \).
4. Determine the intersection points and calculate the enclosed area using definite integrals.
5. Check if you have the same values as a verification step.

Would you like to go through the steps manually, or do you have any other questions? 

**Tip:** Finding the tangent line involves both differentiation and algebraic manipulation—breaking the problem into smaller steps can make the process easier.

Given the function f(x) = x^2 - 2x + 2, answer the following questions:
1. Find the equation of the tangent line to the curve at the point (a, f(a)).
2. Find all values of a such that the tangent line passes through the point P(0, -2).
3. Find the area enclosed by the curve C, the tangent lines, and the point P.

This problem involves finding the tangent line to a quadratic function, determining specific values of a such that the tangent passes through a given point, and calculating the area enclosed by the curve and tangent lines. Ideal for students studying calculus and quadratic functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation