Suppose F is an antiderivative of ƒ and A is an area function of ƒ. What is the relationship between F and A?

Verified step by step guidance

1. Understand the definitions: An antiderivative F of a function ƒ is a function such that the derivative of F is equal to ƒ, i.e., $F\text{'}=ƒ$. An area function A of ƒ represents the accumulated area under the curve of ƒ from a fixed point to a variable point x.
2. Recall the Fundamental Theorem of Calculus: It states that if A(x) is the area function of ƒ, then A'(x) = ƒ(x). This means the derivative of the area function is the original function ƒ.
3. Recognize the connection: Since F is an antiderivative of ƒ, and A'(x) = ƒ(x), it follows that A(x) and F(x) differ by a constant. Specifically, $A\left(x\right)=F\left(x\right)+C$, where C is a constant.
4. Interpret the constant C: The constant C depends on the choice of the lower limit of integration in the area function A(x). If the lower limit is changed, the value of C will adjust accordingly.
5. Summarize the relationship: The area function A(x) is essentially an antiderivative of ƒ, but it includes a constant term that depends on the lower limit of integration. Both F and A are closely related through this constant adjustment.

Key Concepts

Antiderivative

An antiderivative of a function f is another function F such that the derivative of F is equal to f, i.e., F' = f. This means that F represents a family of functions whose slopes at any point correspond to the values of f. Antiderivatives are essential in calculus for solving problems related to integration and finding areas under curves.

Area Function

An area function A associated with a function f typically represents the accumulated area under the curve of f from a specific point to a variable endpoint. Mathematically, it is defined as A(x) = ∫[a to x] f(t) dt, where a is a constant. The area function is crucial for understanding how the total area changes as the endpoint varies, linking it to the concept of integration.

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 establishes that the area function A is directly related to the antiderivative F, as A(x) = F(x) - F(a), illustrating the deep relationship between these concepts.