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Chapter 6: Problem 15

Let \(F(x)\) be an antiderivative of \(f(x)\) (a) If \(\int_{2}^{5} f(x) d x=4\) and \(F(5)=10,\) find \(F(2)\) (b) If \(\int_{0}^{100} f(x) d x=0,\) what is the relationship between \(F(100)\) and \(F(0) ?\)

Short Answer

Expert verified
(a) \(F(2) = 6\); (b) \(F(100) = F(0)\).

Step by step solution

01

Understand the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \(F(x)\) is an antiderivative of \(f(x)\), then \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\). This theorem will be used to solve both parts of this exercise.
02

Solve part (a) using the Theorem

Given \(\int_{2}^{5} f(x) \, dx=4\) and \(F(5)=10\), we must find \(F(2)\). Apply the Fundamental Theorem: \[ F(5) - F(2) = 4 \]. Given \(F(5) = 10\), substitute this into our equation: \(10 - F(2) = 4\). Solve for \(F(2)\): \(F(2) = 6\).
03

Solve part (b) using the Theorem

Given \(\int_{0}^{100} f(x) \, dx = 0\), we want to find the relationship between \(F(100)\) and \(F(0)\). From the theorem, we have \(F(100) - F(0) = 0\). This implies \(F(100) = F(0)\), meaning the antiderivative at these points is the same.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Antiderivative
The antiderivative, sometimes simply called the integral or indefinite integral, is a fundamental concept in calculus. An antiderivative of a function \(f(x)\) is another function \(F(x)\) such that the derivative of \(F(x)\) is equal to \(f(x)\). This is written as \(F'(x) = f(x)\). The process of finding an antiderivative is called "integration."
  • For example, if \(f(x) = 2x\), then an antiderivative, \(F(x)\), is \(x^2\) because the derivative of \(x^2\) is \(2x\).
  • The antiderivative is not unique; it can differ by a constant. This is due to the constant rule of differentiation, where the derivative of a constant is zero. Hence, if \(F(x)\) is an antiderivative of \(f(x)\), then so is \(F(x) + C\), where \(C\) is any constant.
The antiderivative is the key to understanding the idea of accumulation, allowing us to calculate areas, volumes, and other quantities of interest in real-world applications.
Definite Integral
A definite integral is a type of integral that computes the overall change or the total accumulation of a quantity. It is represented as \(\int_{a}^{b} f(x) \, dx\), where \(a\) and \(b\) are the limits of integration. This gives the net area under the curve of \(f(x)\) from \(x = a\) to \(x = b\).
The Fundamental Theorem of Calculus connects differentiation and integration, providing a powerful tool to evaluate definite integrals using antiderivatives:
  • If \(F(x)\) is an antiderivative of \(f(x)\), the definite integral is given by \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\).
  • This means you can calculate the net change in \(F(x)\) over the interval \([a, b]\) by simply taking the antiderivative at the boundaries \(b\) and \(a\) and subtracting \(F(a)\) from \(F(b)\).
Understanding definite integrals is crucial for solving complex real-life problems involving total quantities, such as distance traveled, area, and probability.
Relationship Between Antiderivatives
The relationship between antiderivatives is centered around the idea that they differ by a constant. Given two points, if a function \(f(x)\) has an integral of zero over an interval, then the antiderivative function \(F(x)\) at these points must be equal.
  • In simpler terms, if \(\int_{a}^{b} f(x) \, dx = 0\), then \(F(b) - F(a) = 0\), implying \(F(b) = F(a)\).
  • This outcome tells you something very interesting about the behavior of \(f(x)\): the "accumulated change" over \([a, b]\) is zero; thus, the heights of \(F(x)\) at points \(a\) and \(b\) are the same.
This relationship exemplifies a balance points phenomenon, which is often used in scenarios where you need to determine if two states of a system are equal or unchanged despite any internal fluctuations.

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Most popular questions from this chapter

Are the statements in Problems \(104-112\) true or false? Give an explanation for your answer. An antiderivative of \(1 / x\) is \(\ln |x|+\ln 2\).

Are the statements in Problems \(104-112\) true or false? Give an explanation for your answer. If \(F(x)\) and \(G(x)\) are two antiderivatives of \(f(x)\) for \(-\infty < x < \infty\) and \(F(5) > G(5),\) then \(F(10) > G(10)\).

In Exercises \(66-75,\) decide if the statement is True or False by differentiating the right-hand side. $$\int\left(3 \sqrt{x}+\frac{5}{x^{2}}\right) d x=2 x^{3 / 2}+\frac{5}{x}+c$$

In Exercises \(22-33,\) find the general antiderivative. $$g(x)=\sin x+\cos x$$

In Exercises \(22-33,\) find the general antiderivative. $$p(t)=\frac{1}{\sqrt{t}}$$
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