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Let's work through this problem using the **Fundamental Theorem of Calculus** (FTC), which has two parts:

1. **Part 1 of the Fundamental Theorem of Calculus**: This part states that if \( g(x) = \int_a^x f(t) \, dt \), then \( g'(x) = f(x) \). This is the derivative of the integral function.
   
2. **Part 2 of the Fundamental Theorem of Calculus**: This part states that if \( F(x) \) is an antiderivative of \( f(x) \), then \( \int_a^b f(x) \, dx = F(b) - F(a) \).

Now, let's solve this step by step.

### 1. Use Part 1 of the Fundamental Theorem of Calculus to find \( g'(x) \)

Given:
\[
g(x) = \int_0^x \left( 7 + \sqrt{t} \right) \, dt
\]

Using **Part 1 of the Fundamental Theorem of Calculus**, we differentiate the integral with respect to \( x \). This gives:

\[
g'(x) = 7 + \sqrt{x}
\]

### 2. Use Part 2 of the Fundamental Theorem of Calculus to evaluate \( g(x) \)

We need to evaluate the integral directly:

\[
g(x) = \int_0^x \left( 7 + \sqrt{t} \right) \, dt
\]

This can be split into two integrals:

\[
g(x) = \int_0^x 7 \, dt + \int_0^x \sqrt{t} \, dt
\]

Now, evaluate both integrals:
- The integral of \( 7 \, dt \) is \( 7x \),
- The integral of \( \sqrt{t} \, dt \) is \( \frac{2}{3} t^{3/2} \), so at the limits \( 0 \) to \( x \), it becomes \( \frac{2}{3} x^{3/2} \).

Thus, we get:

\[
g(x) = 7x + \frac{2}{3} x^{3/2}
\]

### 3. Now differentiate your answer for \( g(x) \)

Now we differentiate \( g(x) = 7x + \frac{2}{3} x^{3/2} \) with respect to \( x \):

\[
g'(x) = \frac{d}{dx} \left( 7x + \frac{2}{3} x^{3/2} \right) = 7 + \frac{2}{3} \cdot \frac{3}{2} x^{1/2} = 7 + \sqrt{x}
\]

### Conclusion:

- Using **Part 1** of the Fundamental Theorem of Calculus, \( g'(x) = 7 + \sqrt{x} \).
- Using **Part 2**, we evaluated \( g(x) = 7x + \frac{2}{3} x^{3/2} \), and differentiating this gives \( g'(x) = 7 + \sqrt{x} \).

Do you want more details or have any questions?

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Here are 5 related questions to deepen your understanding:
1. How does the Fundamental Theorem of Calculus relate differentiation and integration?
2. Can you explain why the upper limit of the integral affects the result in Part 1 of the FTC?
3. How would the answer change if the lower limit of the integral were a function of \( x \) rather than 0?
4. What is the significance of the constant when evaluating an indefinite integral?
5. How would you compute \( g(x) \) if the integrand was more complex, like a trigonometric function?

**Tip**: Always check your results by differentiating the antiderivative to ensure consistency.

Suppose g(x) = ∫₀ˣ (7 + √t) dt. Calculate g'(x) in two different ways: 1. Using Part 1 of the Fundamental Theorem of Calculus. 2. Using Part 2 of the Fundamental Theorem of Calculus to evaluate the integral, then differentiate.

Learn how to apply both parts of the Fundamental Theorem of Calculus to calculate g'(x) for the integral ∫₀ˣ (7 + √t) dt. This example covers step-by-step evaluation using differentiation and integration techniques. Suitable for college-level calculus students.

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