Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If $F$ and $G$ are antiderivatives of $f$ on an interval $I$, then $F(x)=G(x)+C$ on $I$.

Name: Problem 97, Chapter 11: Applied Mathematics: For the Managerial, Life, and Social Sciences
Uploaded: 2021-08-16T23:05:41-08:00
Duration: 1 min 22 s
Channel: Stark Ledbetter
Description: Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If F and G are antiderivatives of f on an interval I, then F(x)=G(x)+C on I.

   Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $F$ and $G$ are antiderivatives of $f$ on an interval $I$, then $F(x)=G(x)+C$ on $I$.

Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan 5th Edition

Chapter 11, Problem 97

Instant Answer

Solved by Expert Stark Ledbetter

08/16/2021

Step 1

Step 1: Recall that an antiderivative of a function $f$ is a function $F$ such that $F'(x) = f(x)$ for all $x$ in the interval $I$. Show more…

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Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If $F$ and $G$ are antiderivatives of $f$ on an interval $I$, then $F(x)=G(x)+C$ on $I$.

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Key Concepts

Mean Value Theorem

The Mean Value Theorem guarantees that if a function has a zero derivative over an interval, then it must be constant on that interval. This principle underlies the fact that if two functions have the same derivative, their difference is a function whose derivative is identically zero, implying it is a constant.

Constant of Integration

When computing indefinite integrals, an arbitrary constant (usually denoted as C) is added to account for the fact that all antiderivatives differ only by a constant. This constant ensures that the antiderivative represents the entire family of functions whose derivatives are the same.

Antiderivative

An antiderivative of a function is another function whose derivative equals the original function across the interval. Since differentiation removes constant terms, any two functions that are antiderivatives of the same function can only differ by a constant.

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