# Fundamental Theorem and Maxima

• flyingpig
In summary, the Fundamental Theorem of Calculus is a fundamental concept in calculus that establishes the relationship between derivatives and integrals. It can be used to find the maximum value of a function and has many practical applications in the real world. Some common mistakes when applying it include forgetting the constant of integration and using incorrect limits of integration. Additionally, it only applies to continuous functions.

## Homework Statement

$$Si(x) = \int_{0}^{x}{\frac{sin(t)}{t} dt$$

At what values of x does this function have a local maximum?
2.Solutions

[PLAIN]http://img833.imageshack.us/img833/701/27444263.png [Broken]

## The Attempt at a Solution

So I took the derivative and applying FTC and I got sin(x)/x = 0

sin(x) = 0

$$x=n\pi$$

Then I drew a number line and I got lost. I looked at the solutions and I was even more lost. This is due tomorrow please help

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PF is confused too huh? lol

## What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a fundamental concept in calculus that establishes the relationship between derivatives and integrals. It states that the integral of a function can be calculated by evaluating its antiderivative at the upper and lower limits of the integral.

## How does the Fundamental Theorem of Calculus relate to the concept of maxima?

The Fundamental Theorem of Calculus is closely related to the concept of maxima. In fact, the Fundamental Theorem of Calculus can be used to find the maximum value of a function, as it allows us to calculate the area under the curve of a function. The maximum value of a function occurs at the point where the slope of the tangent line is zero, which is also known as the critical point.

## What is the difference between local maxima and global maxima?

A local maximum is a point on a function where the value of the function is greater than all other points in a small interval around it. On the other hand, a global maximum is the highest point on the entire function, meaning that the function has no higher value anywhere else.

## How can the Fundamental Theorem of Calculus be applied in real-world situations?

The Fundamental Theorem of Calculus has many practical applications in the real world, particularly in areas such as physics, economics, and engineering. It can be used to calculate the area under a curve, which is important in determining quantities such as displacement, velocity, and acceleration. It can also be used to calculate total profit or loss in economics, or to find the maximum volume of a container in engineering.

## What are some common mistakes to avoid when applying the Fundamental Theorem of Calculus?

One common mistake when applying the Fundamental Theorem of Calculus is forgetting to take into account the constant of integration when evaluating integrals. Another mistake is using the wrong limits of integration, which can lead to incorrect results. It is also important to remember that the Fundamental Theorem of Calculus only applies to continuous functions, so it cannot be used for discontinuous functions or functions with infinite discontinuities.