Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.
(c) Evaluate H '(2) .

Verified step by step guidance

1. Step 1: Recognize that the function H(𝓍) is defined as an integral, H(𝓍) = ∫₀ˣ √(4 − t²) dt. To find H'(𝓍), we use the Fundamental Theorem of Calculus, which states that if F(𝓍) = ∫ₐˣ f(t) dt, then F'(𝓍) = f(𝓍), provided f is continuous.
2. Step 2: Apply the Fundamental Theorem of Calculus to H(𝓍). This gives H'(𝓍) = √(4 − 𝓍²), because the integrand √(4 − t²) is continuous for the given domain.
3. Step 3: Substitute 𝓍 = 2 into the derivative H'(𝓍). This means H'(2) = √(4 − 2²).
4. Step 4: Simplify the expression inside the square root. Compute 4 − 2², which simplifies to 4 − 4.
5. Step 5: Conclude that H'(2) = √(0). The derivative at this point is determined by evaluating the square root of the simplified expression.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then the integral of f from a to b can be computed as F(b) - F(a). This theorem also implies that if H(x) is defined as an integral of a function, then H'(x) can be found by evaluating the integrand at the upper limit of integration.

Differentiation of an Integral Function

When differentiating a function defined by an integral, such as H(x) = ∫₀ˣ f(t) dt, the derivative H'(x) can be computed using the integrand evaluated at the upper limit. Specifically, H'(x) = f(x), provided that f is continuous on the interval. This principle simplifies the process of finding derivatives of integral-defined functions.

Evaluating the Integrand

To evaluate H'(2) in the given problem, we first need to identify the integrand, which is √(4 - t²). We then substitute the upper limit of integration, x = 2, into the integrand. This step is crucial as it allows us to find the value of the derivative at that specific point, which is essential for solving the problem.