Consider the equation x sin x = 5, or in the notation above x sin x – 5 = 0. 

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Instructions
Consider the equation x sin x = 5, or in the notation above x sin x – 5 = 0.  
For this situation, what is f'(x)?
Set up an Excel spreadsheet, or a program in your favorite  programming language, to find solutions to the above equation.  (Submit  this file along with your answers to the questions.)
Starting with a guess of x = 5, what solution does Newton’s method converge to?  (Run the method until the x values don’t change in the first five decimal places.)
There are five solutions to x sin x = 5 between x = 0 and x = 20 (it is at heart a sine wave, after all).  By varying your initial guess, find them all to within five decimal places.

These simulations were tested  utilizing the directional derivative and more traditional stochastic  testing methods and the level of error found between the two on the  location of buffers was found to be negligible.

 
Using a simulation of the manufacturing process Mr. Gürkan found a  method to find the optimal location to place buffers to help machines  run smoother and longer without breaking down. The simulation ran a  maximum of 50 machines and took data on cycle times and downtime to set  up the production process. Then using the data collected and forming a  directional derivative could be used to infer the approximate location  where the process became too loud or uneven. These points were isolated  and selected for the optimal location for the buffers. Once added into  the simulation the productivity of the process was found to increase as  was the time between break down on the production line.
               To further test the simulation additional production  line simulations were produced. These additional simulations were made  with fewer machines used in the process.  These simulations were tested  utilizing the directional derivative and more traditional stochastic  testing methods and the level of error found between the two on the  location of buffers was found to be negligible.
References
Gürkan, G. (2000). Simulation optimization of buffer allocations in  production lines with unreliable machines. Annals of Operations  Research, 93(1–4), 117–216.  https://libproxy.ecpi.edu:2111/10.1023/a:1018900729338

For example, if you are using this  equation and are given the values (5-3)/(-2-1).

 
There are many types of derivatives used in real life engineering  situations but, the one I am going to talk about is tangents used in  daily life. I specifically want to explain an example of a secant  tangent line used in daily life for engineers. A secant line is a line  that connects to two points on a circle. Astronauts use this to find  things like the distance from the moon while it is orbiting to many  different locations on earth. The formula used for this real life  equation is y-b=[(d-b)/(c-a)](x-a). For example, if you are using this  equation and are given the values (5-3)/(-2-1). Your equation of the  secant line should look like so y-3=-2/3(x-1) then y=(-2/3)x+2/3+3 with  the solution being y=(-2/3)x+11/3.
Reference:
Application of tangents and normal in real life. Unacademy. (2022, April 19). Retrieved October 19, 2022, from          https://unacademy.com/content/upsc/study-material/mathematics/application-of-tangents-and-normal-in-real-        life/#:~:text=of%20the%20tangent.-,For%20example%2C%20when%20a%20cycle%20travels%20down%20a%20road%2C%20that,and%20turned%20from%20another%20endLinks to an external site..

Derivative feedback tells the  servo that we are over the set speed, and it doesn’t need to run anymore  decreasing current sent to the servo.

 
For my discussion, I choose to talk about how derivatives are used  for servo motor controls. First, we need to discuss what a servo motor  is and what it does. A servo motor is a current and voltage-controlled  electrical motor. This motor works on a closed-loop system through the  commands of a servo controller which uses a feedback device to control  the velocity and position of the servo. A great example of this would be  the cruise control in a car. The servo controller would be the driver  setting cruise control to a set speed which sends a voltage signal and  varying current to the servo which controls the throttle until you get  to a certain speed, then maintains that speed. The feedback device would  be your tachometer telling the servo to lower the current being sent if  you go over the set speed and to raise the current if you go under the  set speed. The servo is using Proportional Integral Derivative or PID  which changes the motor’s output based on the set speed and what the  tachometer reads. The PID algorithm uses Proportional feedback which  tells the servo that it needs to go faster to reach the set speed  increasing current sent to the servo. Derivative feedback tells the  servo that we are over the set speed, and it doesn’t need to run anymore  decreasing current sent to the servo. Integral feedback holds the  current at its set amps and holds the position of the servo to keep the  set speed without any outside interactions that would call for the need  of the other two. Below is an image of how the PID algorithm works.
(Collins, 2022)
 When put into an equation it will look something like this:
Apmonitor.com (n.d.)
Thank you for reading.
References:
Collins, D. (2022, October 17). FAQ: What are servo feedback gains, overshoot limits, and position error limits? Motion Control Tips. https://www.motioncontroltips.com/faq-what-are-servo-feedback-gains-overshoot-limits-position-error-limits/Links to an external site.
Proportional Integral Derivative (PID). (n.d.). https://apmonitor.com/pdc/index.php/Main/ProportionalIntegralDerivativeLinks to an external site.

Do not  use a problem that has a full solution from a resource. 

 
You’ll learn the mechanics of integration.  Integration is the opposite of differentiation;  it will allow you to solve the differential equations that model the  mechanical loads on a beam and the charge on a capacitor, and help you  design both skyscrapers and microcircuitry. 
Just as there are rules for differentiating functions, there are corresponding rules for integrating functions; and you’ll become familiar with these integration techniques by presenting them to your classmates.
 
Find a resource with a tutorial/information that will help you  create a specific, original, math problem related to your topic.   Remember, you are NOT teaching the whole topic, but just how to solve  ONE PROBLEM You’ve created.  You must cite your source in APA format.   (Even if you know how to create and solve a problem related to your  topic, you still must include a source that your classmates can  reference to read more about what you are teaching.)
Tell us the problem you’ve created.  Remember, your problem must be ORIGINAL: made up by YOU. DO NOT  use a problem that has a full solution from a resource. 
Tell  us how to solve your problem step by step.  Your solution should  include all mathematical steps as well as explanations in your own  words.  You can type this with the equation editor right here in the  reply box, you can use an image file to show hand-written work, or you  can make a video using the media tool to provide your explanations! 

 for an opportunity for a reply credit, you can research the notations  and tell us what you found about the origin of each of the notations. 

  the notation for a derivative can have many forms like f ‘ (x), y’, or dy/dx.   For an opportunity for a reply credit, you can research the notations  and tell us what you found about the origin of each of the notations.   Give a description of what you found in your own words and give the  references you use in APA format. 

 show all of your  work! 

 Create a problem similar to the one your  classmate has created in their original post.  Solve the problem you  created, following your classmate’s instructions.  Show ALL of your  work! 
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3 Friends went to a shop and purchased 3 toys. Each person paid Rs.10 which is t

3 Friends went to a shop and purchased 3 toys. Each person paid Rs.10 which is the cost of one toy. So, they paid Rs.30 i.e. total amount. The shop owner gave a discount of Rs.5 on the total purchase of 3 toys for Rs.30. Then, among Rs.5, Each person has taken Rs.1 and remaining Rs.2 given to the beggar beside the shop. Now, the effective amount paid by each person is Rs.9 and the amount given to the beggar is Rs.2. So, the total effective amount paid is 9*3 = 27 and the amount given to beggar is Rs.2, thus the total is Rs.29. Where has the other Rs.1 gone from the original Rs.30? I’me in dire need of  Math Homework Help , please solve this for me asap.