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Consider the following initial-value problem:

Given:
\[ f'(x) = \frac{x}{7} \]
with the initial condition \( f(16) = 57 \).

1. Integrate the function \( f'(x) \). (Remember the constant of integration.)
\[
\int f'(x) \, dx = \int \frac{x}{7} \, dx = \frac{1}{14} x^2 + C
\]

2. Find the value of \( C \) using the condition \( f(16) = 57 \).
\[
f(16) = \frac{1}{14} (16)^2 + C = 57
\]

3. Solve for \( C \).

4. State the function \( f(x) \) found by solving the given initial-value problem:
\[
f(x) = \frac{1}{14} x^2 + C
\]

Answer :

1. We integrated f'(x) to obtain ∫f'(x)dx = (1/8) x^8 + C, where C is the constant of integration.

2. By using the condition f(16) = 57, we determined the value of C to be 57 - (1/8)(16^8).

3. The function f(x) is given by f(x) = (1/8) x^8 + (57 - (1/8)(16^8)). This function satisfies the initial-value problem.

To solve the initial-value problem, we need to find the function f(x) by integrating f'(x) and using the condition f(16) = 57.

1. Integration of f'(x):
∫f'(x)dx = ∫x^7 dx
Using the power rule of integration, we increase the power by 1 and divide by the new power:
= (1/8) x^8 + C
Here, C represents the constant of integration.

2. Finding the value of C:
We are given that f(16) = 57. Substituting this into the integrated function:
(1/8)(16^8) + C = 57
We solve this equation for C:
C = 57 - (1/8)(16^8)

3. State the function f(x):
Now that we have found the value of C, we can write the function f(x):
f(x) = (1/8) x^8 + (57 - (1/8)(16^8))

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