Problem 1 - Calculus Early Transcendentals

Question

Fill in the blanks with either of the words the derivative or an antiderivative: If \(F^{\prime}(x)=f(x),\) then \(f\) is _______of \(F\) and \(F\) is ____________ of \(f\).

Step-by-Step Solution

Answer

If \(F^{\prime}(x)=f(x),\) then \(f\) is the derivative of \(F\) and \(F\) is an antiderivative of \(f\).

Step 1: Identify the derivative

Given that \(F^{\prime}(x) = f(x)\), we see that \(f(x)\) is the derivative of the function \(F(x)\). This means, if we differentiate \(F(x)\), we will obtain \(f(x)\).

Step 2: Identify the antiderivative

Now, if \(f(x)\) is the derivative of the function \(F(x)\), then \(F(x)\) can be considered an antiderivative of \(f(x)\). This means, if we integrate \(f(x)\), we will obtain a function similar to \(F(x)\) (up to a constant).

Step 3: Complete the sentence

With the above information, we can now fill in the blanks of the given sentence: "If \(F^{\prime}(x)=f(x),\) then \(f\) is the derivative of \(F\) and \(F\) is an antiderivative of \(f\)."

Key Concepts

Derivative

A derivative of a function represents the rate at which the function's value is changing at any given point. It's like looking at how fast you're going on a speedometer. When we talk about the derivative, we often use the notation \( F'(x) \) or \( \frac{dF}{dx} \).
To find the derivative of a function, we differentiate it. Differentiation is a process that applies rules, like the power rule or the product rule, to find slopes of tangent lines. These slopes tell us how steep the function is at any particular point.

• For example, the derivative of \( x^2 \) is \( 2x \). This means at any point \( x \), the slope of the tangent line is \( 2x \).
• If the function is constant like \( y=5 \), the derivative is 0, indicating no change.

Antiderivative

An antiderivative is a function that reverses the process of differentiation. In simpler terms, if you know the derivative, you can find the original function by integrating. Whereas differentiation gives you the rate of change, finding the antiderivative gives you the accumulated value.
When the exercise states that \( f(x) \) is the derivative of \( F(x) \), it implies that \( F(x) \) is an antiderivative of \( f(x) \). If \( f(x) \) is integrated, the antiderivative could be \( F(x)+C \), where \( C \) represents any constant.

• For instance, if \( f(x) = 2x \), an antiderivative could be \( x^2 + C \).
• This constant \( C \) represents an infinite family of functions that "could have been" differentiated to give the same \( f(x) \).

Integration

Integration is the reverse process of differentiation, used to find an antiderivative or to calculate the area under a curve. When you integrate \( f(x) \), you are summing infinitely small quantities to find a total amount. Think of it as putting together tiny pieces of a puzzle to see the whole picture.
The process of integration can be indefinite or definite.

• Indefinite integration, like finding an antiderivative, doesn't give a specific value but a function with a constant \( C \) as we discussed earlier.
• Definite integration calculates the area under the curve from one point to another and results in a numerical value.

For example, the indefinite integral of \( 2x \) with respect to \( x \) is \( x^2 + C \). A definite integral of the same function from \( x = 0 \) to \( x = 2 \) would be \( \int_{0}^{2} 2x \, dx \), which equals 4 after solving. This process helps in solving many real-world problems, such as finding displacement from velocity.