consider the funciton f(x)=2x^2-5x+5,a. find the linear approximation, L(x) at x=3. b. use your function from part b to approximate f(3.2)

To solve this problem, we need to follow these steps:

### Part (a): Find the linear approximation \( L(x) \) at \( x = 3 \)

The linear approximation of a function \( f(x) \) at a point \( x = a \) is given by:
\[ L(x) = f(a) + f'(a)(x - a) \]

First, we need to find \( f(3) \):
\[ f(x) = 2x^2 - 5x + 5 \]
\[ f(3) = 2(3)^2 - 5(3) + 5 \]
\[ f(3) = 2(9) - 15 + 5 \]
\[ f(3) = 18 - 15 + 5 = 8 \]

Next, we need to find the derivative \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(2x^2 - 5x + 5) \]
\[ f'(x) = 4x - 5 \]

Then, find \( f'(3) \):
\[ f'(3) = 4(3) - 5 \]
\[ f'(3) = 12 - 5 = 7 \]

Now, we can write the linear approximation \( L(x) \) at \( x = 3 \):
\[ L(x) = f(3) + f'(3)(x - 3) \]
\[ L(x) = 8 + 7(x - 3) \]
\[ L(x) = 8 + 7x - 21 \]
\[ L(x) = 7x - 13 \]

### Part (b): Use the linear approximation to approximate \( f(3.2) \)

Using the linear approximation \( L(x) = 7x - 13 \):
\[ L(3.2) = 7(3.2) - 13 \]
\[ L(3.2) = 22.4 - 13 \]
\[ L(3.2) = 9.4 \]

Therefore, the approximation of \( f(3.2) \) using the linear approximation is 9.4.

Would you like further details or have any questions?

Here are some related questions you might be interested in:
1. How do you find the second derivative of a function?
2. What is the purpose of linear approximation in calculus?
3. Can you explain the concept of Taylor series?
4. How do you determine the concavity of a function?
5. What are some real-world applications of linear approximations?

**Tip:** When approximating functions, the closer your point of approximation is to the given point, the more accurate your approximation will be.

Learn how to find the linear approximation of the function f(x) = 2x^2 - 5x + 5 at x = 3 and use it to approximate f(3.2). This problem involves calculus concepts such as linear approximation and derivatives, suitable for advanced high school students.

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