Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
exercises, use the Fundamental Theorem of Calculus, Part 1, to find each derivative.
x
(d/dx) ∫ e^(cos t) dt
1
![# The Fundamental Theorem of Calculus
## Introduction
Before we delve into the proof, a couple of subtleties are worth mentioning here. First, a comment on the notation. Note that we have defined a function, \( F(x) \), as the definite integral of another function, \( f(t) \), from the point \( a \) to the point \( x \). At first glance, this is confusing, because we have said several times that a definite integral is a number, and here it looks like it’s a function. The key here is to notice that for any particular value of \( x \), the definite integral is a number. So the function \( F(x) \) returns a number (the value of the definite integral) for each value of \( x \).
Second, it is worth commenting on some of the key implications of this theorem. There is a reason it is called the *Fundamental Theorem of Calculus*. Not only does it establish a relationship between integration and differentiation, but also it guarantees that any integrable function has an antiderivative. Specifically, it guarantees that any continuous function has an antiderivative.
## Proof
Applying the definition of the derivative, we have:
\[ F'(x) = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h} \]
\[ = \lim_{h \to 0} \frac{1}{h} \left[ \int_{a}^{x+h} f(t) \, dt - \int_{a}^{x} f(t) \, dt \right] \]
\[ = \lim_{h \to 0} \frac{1}{h} \left[ \int_{x}^{x+h} f(t) \, dt + \int_{a}^{x} f(t) \, dt - \int_{a}^{x} f(t) \, dt \right] \]
\[ = \lim_{h \to 0} \frac{1}{h} \left[ \int_{x}^{x+h} f(t) \, dt \right] \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d6a2302-be00-405d-bbf6-4b5c194aa6b1%2Fe7d9cec5-f7f6-4afa-83a8-9df53b45ddc6%2F346v9cu_processed.png&w=3840&q=75)
![# 5.3 The Fundamental Theorem of Calculus
The image is part of a webpage detailing the Fundamental Theorem of Calculus. The section addresses the relationship between a function and its integral using mathematical expressions and logical reasoning.
### Mathematical Expressions
1. The text manipulates the limit of an integral as follows:
\[
\lim_{h \to 0} \frac{1}{h} \left[ \int_{x}^{x+h} f(t) \, dt + \int_{x}^{a} f(t) \, dt \right] = \lim_{h \to 0} \frac{1}{h} \int_{x}^{x+h} f(t) \, dt
\]
2. The simplification of this expression shows:
\[
\lim_{h \to 0} \frac{1}{h} \int_{x}^{x+h} f(t) \, dt = f(x)
\]
3. Analyzing the expression, it is identified that:
\[
\frac{1}{h} \int_{x}^{x+h} f(t) \, dt
\]
represents the average value of \( f(x) \) over the interval \([x, x + h]\).
4. By applying the Mean Value Theorem for Integrals, a number \( c \) exists in \([x, x + h]\) for which:
\[
\frac{1}{h} \int_{x}^{x+h} f(t) \, dt = f(c)
\]
5. Since \( c \) lies between \( x \) and \( x + h \), and as \( h \to 0 \), \( c \) approaches \( x \). Given that \( f(x) \) is continuous:
\[
\lim_{c \to x} f(c) = f(x)
\]
6. The combination of these elements leads to the expression:
\[
F'(x) = \lim_{h \to 0} \frac{1}{h} \int_{x}^{x+h} f(t) \, dt = \lim_{h \to 0} f(c) = f(x)
\]
### Conclusion
The proof concludes that the derivative \( F'(x) \) equates to \( f](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d6a2302-be00-405d-bbf6-4b5c194aa6b1%2Fe7d9cec5-f7f6-4afa-83a8-9df53b45ddc6%2Fowi7qx_processed.png&w=3840&q=75)
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