Problem 2 - Calculus Early Transcendentals

Question

Suppose \(F\) is an antiderivative of \(f\) and \(A\) is an area function of \(f\) What is the relationship between \(F\) and \(A ?\)

Step-by-Step Solution

Answer

Answer: The relationship between an antiderivative function, F, and an area function, A, of a given function f is that the antiderivative function F is equal to the derivative of the area function A with respect to x. Mathematically, this can be represented as: $$F(x) = A'(x)$$

Step 1: Define the Antiderivative Function (F)

An antiderivative function, F, is a function such that its derivative is equal to the original function f. Mathematically, this can be represented as: $$F'(x) = f(x)$$

Step 2: Define the Area Function (A)

An area function, A, is a function that represents the area under the curve of f from a fixed point a (lower bound) to a variable point x (upper bound) on the x-axis. Mathematically, it can be represented as an integral: $$A(x) = \int_{a}^{x} f(t) dt$$

Step 3: Use the Fundamental Theorem of Calculus to Connect F and A

The Fundamental Theorem of Calculus states that: $$\frac{d}{dx}\left(\int_{a}^{x} f(t) dt\right) = f(x)$$

Step 4: Derive the Relationship between F and A

Since \(\frac{d}{dx}(A(x)) = f(x)\), according to the Fundamental Theorem of Calculus, and \(F'(x) = f(x)\) by definition, we can then derive the relationship between F and A as follows: $$\frac{d}{dx}(A(x)) = F'(x)$$ Thus, we can conclude that an antiderivative function F is equal to the derivative of the area function A with respect to x: $$F(x) = A'(x)$$

Key Concepts

Antiderivative

An antiderivative is a function that reverses differentiation. If you have a function \( f(x) \), an antiderivative \( F(x) \) is one that, when differentiated, gives back \( f(x) \). This means that the derivative of \( F(x) \) is \( f(x) \), represented mathematically by:

• \( F'(x) = f(x) \)

Knowing an antiderivative is useful because it allows us to "undo" the derivative and find the original function from its rate of change. There are often many antiderivatives for a single function, differing by a constant term. This constant is generally denoted as \( C \), so a general expression of an antiderivative could be \( F(x) = \int f(x) \, dx = F(x) + C \). Understanding antiderivatives is a key step in solving problems involving integration.

Area Function

The area function, commonly denoted \( A(x) \), is pivotal in calculus as it helps quantify the area under a curve of a function \( f \) over a particular interval. This function is essentially a definite integral, starting from a fixed point \( a \) to a variable endpoint \( x \). It can be expressed as:

• \( A(x) = \int_{a}^{x} f(t) \, dt \)

Imagine the x-axis as a timeline, with \( a \) as your starting point and \( x \) constantly moving. The area under the curve from \( a \) to \( x \) accumulates as \( x \) changes. The beauty of the area function is its connection to physical problems, like finding distances from a velocity-time graph. Its importance is further magnified by the Fundamental Theorem of Calculus, which relates the area function to antiderivatives.

Integration

Integration is a fundamental concept in calculus, referring to the process of finding an integral. It is often used to find the total accumulation of a quantity, such as area, volume, or other totals from rates of change. There are two main types of integration: definite and indefinite.

Indefinite Integration

This is the process of finding an antiderivative of a function, resulting in a family of functions that include a constant of integration \( C \). It is symbolized as:

• \( \int f(x) \, dx = F(x) + C \)

Definite Integration

Used to compute the exact area under a curve between two limits. This involves the area function we spoke about earlier and looks like:

• \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \)

Integration is not only a mathematical operation but also a way to model real-world situations, like calculating total accumulated quantities over time. It's intricately linked with differentiation via the Fundamental Theorem of Calculus, which highlights how differentiating the integral of a function returns us to the original function.