To solve the given problem, we first need to understand the nested integral functions $f(x)$ and $g(y)$, and then calculate $g''\left(\frac{\pi}{6}\right)$. Let's proceed step by step.

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Definitions:

1. $f(x) = \int_{0}^{3+x^2} \sin(t^2) \, dt$
2. $g(y) = \int_{y}^{7} f(x) \, dx$

We aim to compute $g''\left(\frac{\pi}{6}\right)$.

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Step 1: Compute $g'(y)$ using the Fundamental Theorem of Calculus

Using the derivative rule for integrals, we have: $g'(y) = -f(y)$

Step 2: Compute $g''(y)$

Taking the derivative of $g'(y)$, we get: $g''(y) = -f'(y)$

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Step 3: Compute $f'(x)$

Using the Leibniz rule for differentiating an integral with variable limits: $f(x) = \int_{0}^{3+x^2} \sin(t^2) \, dt$

The derivative is: $f'(x) = \sin((3+x^2)^2) \cdot \frac{d}{dx}(3+x^2) = \sin((3+x^2)^2) \cdot 2x$

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Step 4: Substitute into $g''(y)$

From Step 2, $g''(y) = -f'(y)$. Substituting $f'(y)$ into this, we get: $g''(y) = -\left(\sin((3+y^2)^2) \cdot 2y\right)$

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Step 5: Evaluate $g''\left(\frac{\pi}{6}\right)$

Substitute $y = \frac{\pi}{6}$ into the expression for $g''(y)$: $g''\left(\frac{\pi}{6}\right) = -\left(\sin\left(\left(3 + \left(\frac{\pi}{6}\right)^2\right)^2\right) \cdot 2 \cdot \frac{\pi}{6}\right)$

Simplify step by step:

1. Compute $\left(\frac{\pi}{6}\right)^2 = \frac{\pi^2}{36}$.
2. Add $3 + \frac{\pi^2}{36}$.
3. Square the result.
4. Take the sine of the squared value.
5. Multiply by $-2 \cdot \frac{\pi}{6}$.

Let me calculate this explicitly. math

Define constants

pi = math.pi

Step-by-step calculation

y = pi / 6 y_squared = y2 term1 = 3 + y_squared # (3 + (pi/6)^2) term2 = term12 # Square the result sin_value = math.sin(term2) # Sine of the squared value

Final calculation for g''(pi/6)

g_double_prime = -2 * y * sin_value g_double_primeThe value of $g''\left(\frac{\pi}{6}\right)$ is approximately $1.0077$.

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Questions for deeper understanding:

1. How does the Leibniz rule apply to differentiate functions involving integral limits?
2. Can you describe the behavior of $f(x)$ for very large values of $x$?
3. How does the sine function impact the overall behavior of $g(y)$?
4. Why does the negative sign appear in the derivative of $g(y)$?
5. What is the significance of the squared term $(3+x^2)^2$ in the function $f(x)$?

Tip:

Always use the Fundamental Theorem of Calculus and the Leibniz rule carefully when dealing with nested integrals or functions involving variable limits.