6) r has been graphed on the re plane below. (You can think of it as the cartesian form of the polar graph) Use this graph to graph r on the polar plane. Also, the point A has been labeled on the graph below. Label the corresponding point A on your polar graph. 75% -025 02 125 175 225
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
3. **Convert the Cartesian Coordinates to Polar Coordinates:**
To graph the same function on the polar plane, you need to use the polar coordinates system.
- **Point A:** The coordinates of point A are already given as \((1.5, 1.5)\), where \( \theta = 1.5\) and \(r = 1.5\).
4. **Plotting on the Polar Graph:**
A polar graph's radial lines extend outward from the center point, representing the angle \(\theta\), while the concentric circles represent the radius \(r\).
- Label point A on the polar graph using the coordinates \((r, \theta) = (1.5, 1.5)\). To do this:
1. Find the angle \(\theta = 1.5\) radians on the polar graph.
2. Move outwards from the center along the radial line until you reach a radius \(r = 1.5\).
3. Mark this point and label it as point A.
**Polar Graph Explanation:**
The polar graph shown has:
- Radial lines extending outward that represent angles in radians.
- Concentric circles that represent different radii from the center (origin).
Use this graph to plot the function \(r(\theta)\) and point A accurately.
**Note:** It is crucial to practice plotting points in both Cartesian and Polar coordinates as it helps in understanding the relationship and conversion between different coordinate systems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc93a7fac-dada-46a7-8fba-a69354232c26%2Fa29e97e9-2532-48bc-9daa-dc5f0ce29792%2Fu6daofo.jpeg&w=3840&q=75)
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