Let f be a function such that f(1)=2 and f′(1)=2 . Which of the following is the linearization of f at a=1 ? 
Question 1 Select one: 
y=2x−2 cross out 
y=2x−1 cross out 
y=2x cross out 
y=x−1

The **linearization** of a function \( f(x) \) at \( a = 1 \) is given by the formula for the tangent line:

\[
L(x) = f(a) + f'(a)(x - a)
\]

In this case, \( f(1) = 2 \) and \( f'(1) = 2 \). So, applying the formula:

\[
L(x) = f(1) + f'(1)(x - 1) = 2 + 2(x - 1)
\]

Simplifying:

\[
L(x) = 2 + 2x - 2 = 2x
\]

Thus, the linearization of \( f \) at \( a = 1 \) is:

\[
y = 2x
\]

So, the correct answer is:

**y = 2x**

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. How do you find the derivative of a function?
2. What is the difference between linearization and approximation?
3. Can you explain how the linearization works for \( a \neq 1 \)?
4. What is the geometric interpretation of a function’s derivative?
5. How would the linearization change if \( f'(1) \) were different?

**Tip:** Linearization is especially useful for approximating a function near a given point, particularly when the function is too complex to compute directly.

Let f be a function such that f(1)=2 and f′(1)=2 . Which of the following is the linearization of f at a=1 ? Question 1 Select one: y=2x−2 cross out y=2x−1 cross out y=2x cross out y=x−1

Explore the linearization of a function at a given point, including the use of derivatives and the tangent line formula. This example demonstrates how to find the linear approximation for f(1) = 2 and f'(1) = 2. Suitable for students in Grades 11-12 studying Calculus.

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