If x is so small that `x^(3)` and higher powers of x may be neglected, then `((1+x)^(3//2)-(1+1/2 x)^3)/((1-x)^(1//2))` may be approximated as

Options

A. `x/2 - 3/8 x^(2)`
B. `- (3)/(8)x^(2)`
C. `3x+3/8 x^(2)`
D. `1- 3/8 x^(2)`

Answer

To solve the problem, we need to simplify the expression

\[ \frac{(1+x)^{3/2} - (1 + \frac{1}{2} x)^3}{(1-x)^{1/2}} \]

given that \( x \) is small enough that \( x^3 \) and higher powers can be neglected.

### Step 1: Expand \( (1+x)^{3/2} \)

Using the Binomial Theorem, we can expand \( (1+x)^{3/2} \):

\[ (1+x)^{3/2} \approx 1 + \frac{3}{2}x + \frac{3}{8}x^2 \]

### Step 2: Expand \( (1 + \frac{1}{2} x)^3 \)

Next, we expand \( (1 + \frac{1}{2} x)^3 \):

\[ (1 + \frac{1}{2} x)^3 = 1 + 3 \cdot \frac{1}{2} x + 3 \cdot \left(\frac{1}{2} x\right)^2 = 1 + \frac{3}{2} x + \frac{3}{4} x^2 \]

### Step 3: Subtract the two expansions

Now we subtract the two results:

\[ (1+x)^{3/2} - (1 + \frac{1}{2} x)^3 \approx \left(1 + \frac{3}{2}x + \frac{3}{8}x^2\right) - \left(1 + \frac{3}{2}x + \frac{3}{4}x^2\right) \]

This simplifies to:

\[ \frac{3}{8}x^2 - \frac{3}{4}x^2 = \left(\frac{3}{8} - \frac{6}{8}\right)x^2 = -\frac{3}{8}x^2 \]

### Step 4: Expand \( (1-x)^{1/2} \)

Now we expand \( (1-x)^{1/2} \):

\[ (1-x)^{1/2} \approx 1 - \frac{1}{2}x - \frac{1}{8}x^2 \]

### Step 5: Form the final expression

Now we can substitute our results back into the original expression:

\[ \frac{-\frac{3}{8}x^2}{1 - \frac{1}{2}x - \frac{1}{8}x^2} \]

### Step 6: Approximate the denominator

Since \( x \) is small, we can approximate the denominator:

\[ 1 - \frac{1}{2}x - \frac{1}{8}x^2 \approx 1 \]

### Step 7: Final simplification

Thus, we have:

\[ \frac{-\frac{3}{8}x^2}{1} = -\frac{3}{8}x^2 \]

### Conclusion

The final approximation of the expression is:

\[ -\frac{3}{8}x^2 \]

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