Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.
(b) Given an area function A(𝓍) = ∫ₐˣ ƒ(t) dt and an antiderivative F of ƒ, it follows that A'(𝓍) = F(𝓍) .

Verified step by step guidance

1. Step 1: Begin by understanding the problem. The area function A(𝓍) = ∫ₐˣ ƒ(t) dt represents the accumulated area under the curve of ƒ(t) from a fixed point 'a' to a variable point '𝓍'. The goal is to determine whether A'(𝓍) = F(𝓍), where F is an antiderivative of ƒ.
2. Step 2: Recall the Fundamental Theorem of Calculus, Part 1. It states that if ƒ is continuous on [a, 𝓍], then the derivative of the area function A(𝓍) with respect to 𝓍 is equal to the value of the integrand at 𝓍. Mathematically, A'(𝓍) = ƒ(𝓍).
3. Step 3: Understand the relationship between an antiderivative and the integrand. An antiderivative F of ƒ satisfies F'(𝓍) = ƒ(𝓍). This means that F is a function whose derivative is ƒ.
4. Step 4: Compare A'(𝓍) = ƒ(𝓍) (from the Fundamental Theorem of Calculus) with the statement A'(𝓍) = F(𝓍). Since F'(𝓍) = ƒ(𝓍), the statement A'(𝓍) = F(𝓍) is incorrect unless F(𝓍) = ƒ(𝓍), which is not generally true. A'(𝓍) equals ƒ(𝓍), not F(𝓍).
5. Step 5: Provide a counterexample to clarify. Consider ƒ(𝓍) = 𝓍². The area function A(𝓍) = ∫ₐˣ 𝓉² dt has a derivative A'(𝓍) = 𝓍². However, an antiderivative F(𝓍) of ƒ(𝓍) could be F(𝓍) = (1/3)𝓍³ + C, which is not equal to ƒ(𝓍). This demonstrates that the statement A'(𝓍) = F(𝓍) is false.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F is an antiderivative of a continuous function f on an interval [a, b], then the integral of f from a to x is given by A(x) = ∫ₐˣ f(t) dt = F(x) - F(a). This theorem implies that the derivative of the area function A(x) is equal to the original function f evaluated at x, i.e., A'(x) = f(x).

Area Function

An area function A(x) represents the accumulated area under the curve of a function f from a fixed point a to a variable point x. Mathematically, it is defined as A(x) = ∫ₐˣ f(t) dt. This function is crucial in understanding how the total area changes as x varies, and its derivative A'(x) gives the instantaneous rate of change of this area, which corresponds to the value of the function f at that point.

Antiderivative

An antiderivative F of a function f is a function whose derivative is f, meaning F' = f. Antiderivatives are essential in calculus as they allow us to reverse the process of differentiation. In the context of the area function, if F is an antiderivative of f, then the area function A(x) can be expressed in terms of F, leading to the conclusion that A'(x) = F'(x) = f(x).