Question

In the following exercises, find the Taylor polynomials of degree 2 to approximate the given function centered at the given point.

(a) ( f(x) )

Answer 1

For the function
`\( f(x) = e^x \) and center \( c = 0 \)`, the Taylor polynomial of degree 2 is
`\( P_2(x) = 1 + x + (1)/(2)x^2 \)`, calculated by evaluating function values, derivatives, and applying the Taylor polynomial formula.

Step-by-step explanation:

To find the Taylor polynomial of degree 2 for a given function centered at a specific point, we'll use the formula for the Taylor polynomial:

`\[ P_n(x) = f(c) + f'(c)(x - c) + (f''(c))/(2!)(x - c)^2 \]`

where
`\( f(c) \)` is the function value at the center,
`\( f'(c) \)` is the first derivative at the center,
`\( f''(c) \)` is the second derivative at the center, and so on.

Let's assume we have a function
`\( f(x) \)` and we want to find the Taylor polynomial of degree 2 centered at
`\( c \)`.

Here's a step-by-step guide on how to find the Taylor polynomial:

1. Find the function value
`\( f(c) \)` at the center.

2. Find the first derivative
`\( f'(x) \)`.

3. Evaluate
`\( f'(x) \)` at the center
`\( c \)` to find
`\( f'(c) \)`.

4. Find the second derivative
`\( f''(x) \)`.

5. Evaluate
`\( f''(x) \)` at the center
`\( c \)` to find
`\( f''(c) \)`.

6. Substitute these values into the formula to obtain the Taylor polynomial of degree 2.

Let's go through a specific example. Suppose we have the function
`\( f(x) = e^x \)` and we want the Taylor polynomial of degree 2 centered at
`\( c = 0 \)`.

1. Find
`\( f(0) \)`:

`\[ f(0) = e^0 = 1 \]`

2. Find
`\( f'(x) \):`

`\[ f'(x) = e^x \]`

3. Evaluate
`\( f'(x) \) at \( c = 0 \) to find \( f'(0) \):`

`\[ f'(0) = e^0 = 1 \]`

4. Find
`\( f''(x) \)`:

`\[ f''(x) = e^x \]`

5. Evaluate
`\( f''(x) \) at \( c = 0 \) to find \( f''(0) \):`

`\[ f''(0) = e^0 = 1 \]`

6. Substitute these values into the Taylor polynomial formula:

`\[ P_2(x) = 1 + 1(x - 0) + (1)/(2!)(x - 0)^2 \]`

`\[ P_2(x) = 1 + x + (1)/(2)x^2 \]`

So, the Taylor polynomial of degree 2 for
`\( f(x) = e^x \)` centered at
`\( c = 0 \) is \( P_2(x) = 1 + x + (1)/(2)x^2 \).`