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Chapter 5: Problem 1

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

Short Answer

Expert verified
Answer: The relationship between the area function, A, and the function f is given by A'(x) = f(x), where A'(x) denotes the derivative of A(x) with respect to x.

Step by step solution

01

Understanding the concept of area functions

An area function, denoted as \(A(x)\), calculates the signed area under the curve of a continuous function, \(f(x)\), over the range \([a, x]\), where \(a\) is a constant and \(x\) is a variable. The relationship between the area function and the original function can be found using the Fundamental Theorem of Calculus.
02

Using the Fundamental Theorem of Calculus (FTC)

The Fundamental Theorem of Calculus states that if \(F(x)\) is an antiderivative of a continuous function \(f(x)\) on the interval \([a, x]\), then \[\int_a^x f(t) dt = F(x) - F(a).\] In this case, \(A(x)\) is an antiderivative of \(f(x)\) (by definition of an area function). Thus, we can apply the FTC to find the relationship between \(f\) and \(A\).
03

Applying FTC to find the relationship

Since \(A(x)\) is an antiderivative of \(f(x)\) on the interval \([a, x]\), we have \[A(x) - A(a) = \int_a^x f(t) dt.\] We also know that \(A'(x) = f(x)\), as A(x) is an antiderivative of f(x). Hence, the relationship between \(A\) and \(f\) is: \[A'(x) = f(x).\]

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Most popular questions from this chapter

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Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\) and interpret the results using the graphs of part \((b)\) $$f(x)=\cos \pi x ; a=0, b=\frac{1}{2}, c=1$$
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