Define a function A byA(x) =∫ x1√t dtfor x ≥ 1.i Draw a diagram that indicates the meaning of A(x).ii. What does the Fundamental Theorem of Calculus say about thefunction A?iii. What is A′(4)?

i. The function A(x) is defined as the area under the curve of the function f(t) = √t from 1 to x, for x ≥ 1. This can be represented by a diagram showing the curve y = √t, a vertical line at t = 1, another vertical line at t = x, and the area between the curve and the x-axis, bounded by these two vertical lines.

ii. The Fundamental Theorem of Calculus states that if a function f is continuous over the interval [a, b] and F is an antiderivative of f on [a, b], then the definite integral of f from a to b is equal to F(b) - F(a). In the case of the function A(x), the function being integrated, √t, is continuous for t ≥ 0, and the limits of integration are 1 and x. Therefore, A(x) is the antiderivative of √t evaluated at x and 1, or A(x) = 2/3 * (x^(3/2) - 1^(3/2)).

iii. A′(x) is the derivative of A with respect to x. According to the Fundamental Theorem of Calculus, the derivative of a function defined by an integral like A(x) is simply the original function being integrated. Therefore, A′(x) = √x. So, A′(4) = √4 = 2.