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.
In the graph above:
- The \( x \)-axis and \( y \)-axis are labeled.
- The function has a complex, non-linear curve starting from the origin and extending towards positive \( x \)-values.
#### Completing for Even Functions
To create an even function, the graph must be symmetric with respect to the \( y \)-axis. This means the left side should mirror the right side. The diagrams below show different potential completions of the graph for \( x < 0 \):
1. **First Row, First Column:**
- The left side is a mirror image of the right, showing perfect symmetry.
2. **First Row, Second Column:**
- The left side partially mirrors the right side but diverges midway.
3. **Second Row, First Column:**
- The left side starts as a partial mirror but diverges significantly near negative \( x \)-values.
4. **Second Row, Second Column:**
- The left side does not mirror the right side properly.
The appropriate completion for an even function will be the first in the list as it shows symmetry.
#### Completing for Odd Functions
To create an odd function, the graph must be symmetric with respect to the origin. This means that for every point \( (x, y) \) in the graph, there should be a corresponding point \((-x, -y)\). The diagrams below illustrate different completions to make the graph an odd function:
1. **First Row, First Column:**
- Both sides show symmetrical properties with respect to the origin.
2. **First Row, Second Column:**
- The symmetry is imperfect; negative \( x \)-values do not properly mirror positive \( x \)-values.
3. **Second Row, First Column:**
- The attempt of symmetry fails as negative and positive \( x \)-values do not line up correctly.
4. **Second Row, Second Column:**
- Shows no proper origin-centred symmetry.
The correct completion for an odd function will be the first in the list, as it effectively demonstrates origin symmetry.
By understanding these principles, you](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F82021bcf-518d-429d-b429-ad31a2321057%2F92faf67f-93ae-4ea1-8ca2-e558be3212ea%2F7aofxmq_processed.png&w=3840&q=75)
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