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To approximate \( e^x \) with a rational function of the form

\[
\frac{a_0 + a_1 x}{1 + b_1 x},
\]

we can use a Taylor series expansion approach to match terms.

### Step 1: Taylor Series Expansion for \( e^x \)

The Taylor series of \( e^x \) around \( x = 0 \) is:

\[
e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \cdots
\]

### Step 2: Expanding the Rational Function

We also expand the rational function \(\frac{a_0 + a_1 x}{1 + b_1 x}\) in a Taylor series around \( x = 0 \). 

To do this, perform a division of \( a_0 + a_1 x \) by \( 1 + b_1 x \):

\[
\frac{a_0 + a_1 x}{1 + b_1 x} = a_0 + (a_1 - a_0 b_1)x + a_0 b_1^2 x^2 + \cdots
\]

### Step 3: Matching Terms

Now we compare the Taylor expansion terms of the rational function with those of \( e^x \). By matching terms up to \( x^2 \), we get:

1. **Constant term**: \( a_0 = 1 \)
2. **Coefficient of \( x \)**: \( a_1 - a_0 b_1 = 1 \)
3. **Coefficient of \( x^2 \)**: \( a_0 b_1^2 = \frac{1}{2} \)

### Step 4: Solving for \( a_0 \), \( a_1 \), and \( b_1 \)

1. From the constant term, \( a_0 = 1 \).
2. Substitute \( a_0 = 1 \) into the coefficient of \( x \) term:
   \[
   a_1 - b_1 = 1 \Rightarrow a_1 = 1 + b_1
   \]
3. Substitute \( a_0 = 1 \) into the coefficient of \( x^2 \) term:
   \[
   b_1^2 = \frac{1}{2} \Rightarrow b_1 = \pm \frac{1}{\sqrt{2}}
   \]

To keep things simple, we usually take the positive solution, so \( b_1 = \frac{1}{\sqrt{2}} \).

4. Substitute \( b_1 = \frac{1}{\sqrt{2}} \) into the equation for \( a_1 \):
   \[
   a_1 = 1 + \frac{1}{\sqrt{2}}
   \]

### Final Answer

The rational approximation of \( e^x \) is

\[
\frac{1 + \left(1 + \frac{1}{\sqrt{2}}\right) x}{1 + \frac{1}{\sqrt{2}} x}.
\]

Would you like further details or have any questions?

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Here are some related questions to explore this concept further:

1. How can we find higher-order rational approximations for \( e^x \)?
2. Why is it common to use rational approximations for exponential functions?
3. How does this rational approximation compare in accuracy with Taylor polynomials?
4. Can we generalize this approach to approximate other functions, like \( \sin(x) \)?
5. What are the practical applications of using rational approximations?

**Tip:** Rational approximations often converge faster than polynomial approximations for functions with complex behavior, making them useful in computational applications.

Obtain the rational approximation of the form \( \frac{a_0 + a_1 x}{1 + b_1 x} \) to \( e^x \).

Learn how to find a rational approximation of the exponential function \( e^x \) using a function of the form \( \frac{a_0 + a_1 x}{1 + b_1 x} \). This solution uses Taylor series expansion to match terms and solve for constants. Suitable for undergraduate students.

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