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?

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.