The anti - derivative of `f(x) = e^(x//2)` whose graph passes through the point (0,3) is :

Options

A. `3e^(x//2)`
B. `4 e^(x//2-1)`
C. `2e^(x//2+1)`
D. None of these

Answer

To find the anti-derivative of the function \( f(x) = e^{x/2} \) that passes through the point \( (0, 3) \), we will follow these steps:

### Step 1: Find the anti-derivative
The anti-derivative (or integral) of \( f(x) = e^{x/2} \) can be calculated using the formula for the integral of an exponential function:

\[ \int e^{ax} \, dx = \frac{e^{ax}}{a} + C \]

In our case, \( a = \frac{1}{2} \).

### Step 2: Apply the formula
Thus, we have:

\[ \int e^{x/2} \, dx = \frac{e^{x/2}}{1/2} + C = 2e^{x/2} + C \]

### Step 3: Use the point (0, 3) to find C
Now, we know that the graph of the anti-derivative passes through the point \( (0, 3) \). This means when \( x = 0 \), \( y = 3 \).

Substituting \( x = 0 \) into our anti-derivative:

\[ y = 2e^{0/2} + C = 2e^{0} + C = 2 \cdot 1 + C = 2 + C \]

Setting this equal to 3 (since the point is \( (0, 3) \)):

\[ 2 + C = 3 \]

### Step 4: Solve for C
Now, we can solve for \( C \):

\[ C = 3 - 2 = 1 \]

### Step 5: Write the final anti-derivative
Now that we have \( C \), we can write the final form of the anti-derivative:

\[ y = 2e^{x/2} + 1 \]

### Final Answer
The anti-derivative of \( f(x) = e^{x/2} \) whose graph passes through the point \( (0, 3) \) is:

\[ y = 2e^{x/2} + 1 \]

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