Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet. Setting it equal to zero and solving for Q we have: Applying the second order condition to ensure whether it is really minimum we take the second derivative of AC function. Differentiation is a procedure used to find a derivative of a function. Let: 6) are on the demand curve. It is available at https://lms.uconn.edu/ultra/courses/_80016_1/cl/outline . For example, efficiency in the use of resources requires that a firm should produce at the minimum possible cost per unit of output. Get subscription and access unlimited live and recorded courses from Indias best educators. equal to zero, both it at maximum and minimum profit levels. Explore the Computer Technology Competencies page for more information.. Students will be given an opportunity to provide feedback on their course experience and instruction using the Universitys standard procedures, which are administered by the Office of Institutional Research and Effectiveness (OIRE). Success in this course program depends heavily on your personal health and well-being. They are used to find many problems like maxima, minima of the function, inflection of the point, the slope of the curve. Use integrals/antiderivatives to solve problems, like finding cost from marginal cost, net change from a rate of change, or the area under or between curves. Differential Calculus has a wide range of applications in business. Models TI-89 and above(including TI-Nspire) are not permitted on the exams. Calculus In Computer Science. Let f(x) be a function, then f(x) = dy/dx is said to be known as a differential equation, where f(x) is known as the derivative of the function, x is known as an independent variable and, y is known as a dependent variable. It will be seen that profit maximisation curve reaches its maximum point at point H. Therefore, at point H, the slope of the tangent (which measures the value of derivative d / dQ) drawn to the profit curve at this point is equal to zero. Differentiation in business refers to the act of marketing a particular product or service in a way that makes it stand out against other products or services. Which one of the following is the correct value of ( 161 162 ) ? Finding the Inflection Points. In fact, they also closely mimic how we find Absolute Extrema! The cookie is used to store the user consent for the cookies in the category "Performance". Thus. 94 b. These cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc. To ensure that the derivative is zero at the profit maximising level of the decision variable (i.e. Please contact your instructor to discuss any needs to you. Similarly, optimum use of resources requires that cost be minimised for producing a given level of output. Consider that a firm is producing two products X and Y. The Mean Value Theorem. Student Athletes and Students with Disabilities should inform your instructor of your commitments as an athlete, any special needs that you have, etc. In this, we will discuss the applications and concept of. Find Where the Function Increases/Decreases. The production costs, in dollars, per week of producing x widgets is given by, In this lecture, we obtain the maximum profit and maximum revenue from this two business problems by using differential calculus Wendel Clark began writing in 2006, with work published in academic journals such as "Babel" and "The Podium." It does not store any personal data. University students are expected to demonstrate competency in Computer Technology. Which of the following has no circle? They are due each Friday. Textbook. WebThe Application of Differential Equations in Business In the following example we shall discuss the application of simple differential equation in business. If you miss an exam, you must contact your course instructor within 48 hours to discuss the possibility of a make-up. They will consist of a mix of multiple choice and free response questions. ~x : EP, Letbe a graph. Content Guidelines 2. We also share information about your use of our site with our social media, advertising and analytics partners who may combine it with other information that youve provided to them or that theyve collected from your use of their services. Access to our library of course-specific study resources, Up to 40 questions to ask our expert tutors, Unlimited access to our textbook solutions and explanations. Setup and solve optimization and related rates problems in various contexts, using derivatives. Since second derivative of AC function is positive, d2 (AC)/ dQ2 > 0, output of 180 units of output is one that minimises average cost of production. Differential calculus is concerned with It is a process by which we find the instant rate of change in a mathematical function based on one of its variables. 1: Summary Marginal analysis techniques help management in determining optimal revenue, profit and cost. This cookie is set by GDPR Cookie Consent plugin. Calculus is used extensively within physics, computer graphics, and in actuarial science. Differentiating the profit function with respect to X while holding Y constant we have, Differentiating the profit function with respect to y while holding X constant we have. See your instuctor for section specific details. It will be seen from this figure that up to point L, marginal profit (d / dQ) that is, slope of the total profit curve, is negative and has been causing the total profits to fall (in fact, up to L, due to negative marginal profits loses have been increasing. Differential calculus also has applications in engineering and economics. Step-by-Step Examples. According to the second order condition, for profit maximisation, the second derivative of the profit function must be negative, that is, d2 / dQ2 < 0. buildings, tunnel, road, and bridge are not safe as they are today. Differentiation and integration can help us solve many types of real-world problems. The average cost of production can be determined using differential calculus. Refer to your individual instructor for additional details. Maximisation of profits through the use of derivative is graphically shown in Figure 5.9. Consumer buying process these are the stages a consumer need to follow while making a decision of bu, Here are the 7 most productive bank customer retention strategies you should try in your business.do, 544 003026074 003027997 Hes done it 206 times 545 003028201 003029874 And Im not, 20 Compare and contrast the five stems of OD laboratory training action, CS 513 Spring 2015 CS 513 Spring 2015 17 17 k k Means Clustering Means, Biochem II Exam 1 (chapter 21 and 25).pdf, the knowledge base the input of the learner module is updated The system, 26 Figure 5 Negative externality and Pollution Source Field Survey 464, Corporate Social Responsibility Campaigns..doc, 47271083_10217313879309563_5951461668679581696_n.jpg, 1 A lease is a contract or part of a contract that conveys the right of use an, b Computers can be used for research activity c Computers can be used for, 1.1.6 Conditional Distributions and Independence.pdf, When Amir opened his seafood market in Alabama he was amazed at the number of, The percentage tax is generally a 3 of sales and receipts b 3 of purchases c 3, Operational Control is the process of assuming that specific tasks are carried, 1. Without calculus, we would not have a clear understanding of the laws that govern our universe. in the power function f(x) = -2x 3 , what is the end behavior of f(x) as x goes to , If algae grows at a rate that can be modeled by the exponential function A(t) = ae rt , what is the surface area of algae on a pond if the initial area of the algae is 2 square meters, the growth. Advertisement cookies are used to provide visitors with relevant ads and marketing campaigns. . All due dates for online homework will be listed in HuskyCT. This expectation is based on the various course activities, assignments, and assessments and the University of Connecticuts policy regarding credit hours. Share Your PPT File. Or your instructor. The Domain of the function can be specified as all the input values in the function, and on the other hand, Range can be defined as all the output values of a function. You also have the option to opt-out of these cookies. It can be used to find solutions to equations, understand the behavior of curves and surfaces, and calculate rates of change. Average Cost. Shortness of breath or difficulty breathing. Derivatives of function, representing cost, strength, materials in a process, profit, etc., are used to determine the About Press Copyright Contact us Creators Advertise Developers Terms Privacy Policy & Safety How YouTube works Test new features Press Copyright Contact us Calculus. Marginals. The derivative of C is also called the marginal cost. Use the tangent line to approximate functions. (More information related to hours per week per credit can be accessed at the Online Student website). Jenn, Founder Calcworkshop, 15+ Years Experience (Licensed & Certified Teacher). All changes will be communicated in an appropriate manner. We strongly encourage you to reframe challenges as an unavoidable pathway to success. Differential calculus is a powerful tool for analyzing mathematical problems. Integration is one of the major parts of calculus, besides differentiation. Discuss five application of differential Calculus in business. Note: We know that this transition to online classes has the chance to increase the number of students who need accommodations and the form those accommodations will take. This is at your instructors discretion. This website includes study notes, research papers, essays, articles and other allied information submitted by visitors like YOU. PMVVY Pradhan Mantri Vaya Vandana Yojana, EPFO Employees Provident Fund Organisation. Evaluate both indefinite and definite integrals, using basic antiderivative rules, including substitution. (More information related to hours per week per credit can be accessed at the, Student Health and Wellness Mental Health, Are you interested in forming a study group with other students in the class? For this reason todays differential calculus was earlier named as infinitesimal calculus. If you do not have UConns student health insurance plan, most major insurance plans are also accepted. Some of the most important applications of differential calculus are optimization problems in which the goal is to find the optimal (best) solution. Sir Isaac Newton used differential calculus to develop his theory of gravity. Marginal cost (MC) is the extra, or additional, cost of producing one more unit of output. There is a study group application in Nexus that can help you get started. This cookie is set by GDPR Cookie Consent plugin. Use mathematical models to represent various business and economics related functions, including cost, revenue, profit, and demand. If they sell x widgets during the year then their profit, in dollars, is Thus the firm will maximise profits of it produces and sells 10.45 units of product X and 8.2 units of product Y1. Here a link to the Mechanical Universe on Differentiation: 1 You discover you have 5 different single socks that are missing their "mate". Application Of Differential Calculus In Physics: The first application that we will look at is physics. To conclude, we get a following general test for maximum and minimum: (1) If the second derivative d2y / dx2 of a function is negative (< 0) at the point where the first derivative (dy / dx) is zero, it will represent a point of maximum. In this article different types of matrices which are important parts of the business and mathematics have been discussed with an explanation. The primary objective of differential calculus is to partition something into smaller parts (infinitesimal parts), in order to determine how it changes. The process of optimisation often requires us to determine the maximum or minimum value of a function. Integral calculus is utilised in several parts such as science, mathematics, engineering, etc. On the other hand, if the second derivative at a point on a profit function where first derivative is zero is positive, it shows profits are in fact minimum rather than maximum. If you do not have UConns student health insurance plan, most major insurance plans are also accepted. Checking if Differentiable Over an Interval. The total cost to the council associated with the campaign can be modeled as. There are several methods for calculations, among which there are differentiation and integration. . The indefinite integrals are used mainly as definite integrals and antiderivatives. It is a very broad and important topic in Higher classes, including integration by parts and substitution. We will discuss the differential and integral calculus concepts in this differential and integral calculus pdf. All students, including students with Deans approval for rescheduled final exams, are required to take the final exam in-person. Differential calculus is also used to determine the optimum amount of fertilizer or pesticide that should be applied on crops to maximize yield and reduce cost. The WMA Bus Lines offers sightseeing tours of Washington, D.C. One tour, priced at $7 per person, had an average demand of about 1000 customers per week. WebApplication of Differential Calculus to Business Problems Revenue Functions In general, a business is concerned not only with its costs, but also with its revenues. If you have difficulty accessing HuskyCT, you have access to the in person/live person support options available during regular business hours through the Help Center. You take the derivative of the position function with respect to time to find out how fast the object is moving. Students who require accommodations should contact the Center for Students with Disabilities, Wilbur Cross Building Room 204, (860) 486-2020 or http://csd.uconn.edu/. Additional information including what to do if you test positive or you are informed through contract tracing that you were in contact with someone who tested positive, and answers to other important questions can be found here: https://studenthealth.uconn.edu/updates-events/coronavirus/, The University of Connecticut is committed to protecting the rights of individuals with disabilities and assuring that the learning environment is accessible. WebDiscuss five application of differential Calculus in business Use in Profit Maximisation: For example, consider the following profit function: = -100+ 160Q-10Q2 Where = profits We have just scratched the surface of what this amazing subject can do. In the exponential function f(x) = 3 -x + 2, what is the end behavior of f(x) as x goes to ? output in the present case), we require to apply the second order condition. Explore the, Students will be given an opportunity to provide feedback on their course experience and instruction using the Universitys standard procedures, which are administered by the, Office of Institutional Research and Effectiveness, Math 1020Q Problem Solving (Spring 2023), Math 1030Q Elementary Discrete Mathematics (Spring 2023), Math 1070Q Mathematics for Business and Economics (Spring 2023), Math 1071Q Calculus for Business and Economics (Spring 2023), Math 2110Q Multivariable Calculus (Spring 2023), Math 2410Q Elementary Differential Equations, Math 2705W Technical Writing in Mathematics, https://www2.math.uconn.edu/ClassHomePages/Math1071/Textbook2. Share Your PDF File Thus, we find that at 8 units of output profits will in fact be maximum. For the optimum value, the first derivative being equal to zero is a necessary condition for maximum or minimum, but it is not a sufficient condition. It is also referred to as a Riemann Integral. The University of Connecticut is committed to protecting the rights of individuals with disabilities and assuring that the learning environment is accessible. Scribd is the world's largest social reading and Astronomy Space flight engineers frequently use calculus when planning lengthy missions. chose answer :(5,10,20,25,100) My solution is: take, can you help me solve this problem? The rate of change of distance covered concerning time in a direction is known as velocity. (See Fig. This cookie is set by GDPR Cookie Consent plugin. Calculus is an in-depth study of functions, and differential calculus studies how fast or slow a function changes. In general, a business is concerned not only with its costs, but also with its revenues. Application of Differential Calculus in Daily Activities, Differential Calculus is a part of mathematics that focuses on dissecting a major, component into little quantities to see how it affects the entire system. Application 1 : Exponential Growth - Population. This website uses cookies to improve your experience while you navigate through the website. Using the pointslope formula for the line through these two points, we have: Create your own unique website with customizable templates. For a function to be a maximum (or minimum) its first derivative is zero. 95 c.96 d.97 e.98, Below is the question to be answered Let the set of all perfect people be called P. What is the correct mathematical notation for the following statement: Nobody is perfect. The difference between the buying and selling price is called profit/loss. Our websites may use cookies to personalize and enhance your experience. In the subject of Mathematics, derivatives have very high utilisation. In this differential calculus and integral calculus pdf, we have discussed the applications of differential integral and calculus. A function can be specified as a relation from the set of inputs to the set of outputs, in which each input is related to one output. For example, a fast-food chain might specialize in healthy foods rather than typical high-fat offerings. The Application of Differential Equations in Business In the following example we shall discuss the application of simple differential equation in business. Forces: The forces acting on a fluid can be calculated using differential calculus. The cookies is used to store the user consent for the cookies in the category "Necessary". Application of Calculus in Commerce and Economics OPTIONAL - II Mathematics for Commerce, Economics and Business 41 APPLICATION OF CALCULUS IN COMMERCE AND ECONOMICS We have learnt in calculus that when 'y' is a function of 'x', the derivative of y w.r.to x i.e. Then, that chain would promote the nutritional value of its menu. In the event of make-up exams, exams proctored using either WebEx or Lockdown Browser with, All due dates for online homework will be listed in HuskyCT. Graphical analysis cannot tell us easily exactly at what level of output, profits will be maximum, for it takes time to draw a graph and conclude from it. This shows that point H at which first derivative, d / dQ is zero and also beyond which second derivative (d2 / dQ2), that is, slope of the first derivative becomes negative is indeed the point of maximum profit. OpenStax Calculus Volume 1: UConn Custom Business Calculus Edition. If P is the principal Now consider point H on the total profit corresponding to output level OQ2. It can be pictured as; Indefinite are the integrals in which the limits are not defined, or the upper and lower limits are not defined. Thus, To solve the above two equations simultaneously we multiply equation (i) by 8 and adding (i) and (ii) we have. Copy and paste text, graphics or hyperlinks. These cookies ensure basic functionalities and security features of the website, anonymously. Student technology training is now available in a new HuskyCT short course created by students for students. There is a study group application in Nexus that can help you get started. Definite integrals have definite limits or definite upper and lower limits. Which of the following is a Fibonacci. The University of Connecticut is dedicated to supporting and enhancing teaching effectiveness and student learning using a variety of methods. A profit function curve such as the one drawn in Figure 5.10 may have both minimum point and maximum points. In this article, we will explore some of the ways in which differential calculus can be applied in real-life situations. Multivariate Optimisation: When a dependent variable is a function of many Reflect on your role in taking care of yourself throughout the semester, before the demands of exams and projects reach their peak. Please do not hesitate to ask questions during class or attend office hours. The instructor reserves the right to change dates accordingly as the semester progresses. We have differentiated the function concerning x. ). This course is an introduction to calculus and applications designed primarily for students majoring in business, economics, or the life and social sciences. d P / d t = k P. where d p / d t is the first derivative of P, k > 0 and t is the time. In this lesson we are going to expand upon our knowledge of derivatives, Extrema, and Optimization by looking at Applications of Differentiation involving Business and Economics, or Applications for Business Calculus. If they sell x widgets during the year then their profit, in dollars, is given by, Science and engineering Differential calculus is used extensively in science and engineering. For example, problems in marketing, economics, inventory analysis, machine learning, and business are all concerned with finding the best solution. the slope of the first derivative) is positive, and since profits will be increasing beyond this point, it cannot be point of maximum profits. various fields such as engineering, medicine, and research analysis. Similarly, in currency trading or foreign exchange, differential calculus can be used to determine the best time for buying and selling currencies. Applications of Differentiation. What is Differential Calculus? In differential calculus basics, you may have learned about differential equations, derivatives, and applications of derivatives. For any given value, the derivative of the function is defined as the rate of change of functions with respect to the given values. Differentiation is a process where we find the derivative of a function. \[C\left( x \right) = 200 + 0.5x + \frac{{10000}}{x}\] However, it is easier to use differential calculus to find the profit-maximising output. There are weekly worksheets designed to give you an idea of what exam questions will look like. He has worked in the field of management and is completing his master's degree in strategic management. It will be seen from Figure 5.10 that point L represents the minimum point and H represents the maximum point of the profit curve. A company can produce a maximum of 1500 widgets in a year. By clicking Accept, you consent to the use of ALL the cookies. How many widgets should they try to sell in order to maximize their profit? (SHaW-MH) office. To keep track of your performance in the course, refer to My Grades in HuskyCT. We also use third-party cookies that help us analyze and understand how you use this website. Use the first and second derivative of a function, as well as its properties like domain, asymptotes and symmetry, to understand the overall shape of a function and draw its graph. Lets say you are a biologist, and you want to calculate how fast a species of bacteria grows over time. Itwillprepare you to use theIT systems and services that youwill use throughoutyourtime at UConn, whether learning online or on-campus. The extra, or additional, cost of producing one More unit of output costs, but also its... Would promote the nutritional value of its menu be accessed at the minimum possible cost per unit output. Hours to discuss the application of differential integral and calculus the use of all cookies... Derivative is zero 1: UConn Custom business calculus Edition this reason differential. Be modeled as please do not have UConns student health insurance plan, most major insurance plans also! The extra, or additional, cost of producing one More unit of output applied in real-life situations fluid... Students, including cost, revenue, profit and cost supporting and enhancing teaching effectiveness and learning! Study notes, research papers, essays, articles and other allied information submitted by like. Example we shall discuss the application of simple differential equation in business unit... Selling currencies procedure used to store the user consent for the cookies in the category Performance! Various business and mathematics have been discussed with an explanation concerning time in a new HuskyCT course! Vandana Yojana, EPFO Employees Provident Fund Organisation a very broad and important topic Higher... The behavior of curves and surfaces, and research analysis are those that are being analyzed and not! You must contact your instructor to discuss any needs to you optimisation often us! Rate of change a variety of methods in fact be maximum to supporting enhancing... Have the option to opt-out of these cookies ensure basic functionalities and security features of ways! Homework will be listed in HuskyCT solve many types of real-world problems solve... Homework will be seen from Figure 5.10 that point L represents the minimum point and maximum points cookies to your. Also called the marginal cost ( MC ) is the correct value of a make-up and.... Our universe his theory of gravity expectation is based on the demand curve in determining optimal revenue, and. Application in Nexus that can help you get application of differential calculus in business final exam in-person and lower limits fact they... Calculus Edition ensure that the learning environment is accessible cost, revenue,,. With relevant ads and marketing campaigns as a Riemann integral change dates accordingly as the progresses. Consist of a function to be a maximum of 1500 widgets in a new HuskyCT short course by. Might specialize in healthy foods rather than typical high-fat offerings to output level.... Your pdf File Thus, we find the derivative of a mix of multiple choice and free response questions in. 1500 widgets in a new HuskyCT short course created by students for students analysis techniques help in. First derivative is zero at the profit maximising level of output a of. Various fields such as engineering, etc weekly worksheets designed to give you an idea of what exam questions look... Cost be minimised for producing a given level of the major parts of the website, anonymously produce a (... Challenges as an unavoidable pathway to success committed to protecting the rights individuals. As engineering, medicine, and you want to calculate how fast the object is moving the semester progresses of... Basic antiderivative rules, including cost, revenue, profit and cost selling currencies or... Personalize and enhance your experience while you navigate through the use of resources requires a. In business Vaya Vandana Yojana, EPFO Employees Provident Fund Organisation referred to as Riemann. Pathway to success and concept of antiderivative rules, including cost, revenue, profit and cost disabilities assuring! Communicated in an appropriate manner might specialize in healthy foods rather than high-fat. A maximum ( or minimum value of its menu Thus, we would not have a clear of. Enhancing teaching effectiveness and student learning using a variety of methods & Teacher! Following example we shall discuss the application of simple differential equation in.! Management and is completing his master 's degree in strategic management in engineering and economics problems. Forces: the forces acting on a fluid can be calculated using differential calculus studies fast! Represents the maximum or minimum value of its menu to zero, both it at maximum and minimum levels. Plans are also accepted point of the position function with respect to time to out... Two points, we will explore some of the website live and recorded courses from best. And is completing his master 's degree in strategic management an exam, you to... The council associated with the campaign can be determined using differential calculus studies how fast the object is.... A maximum of 1500 widgets in a direction is known as velocity the rate of change of distance concerning. Of visitors, bounce rate, traffic source, etc strongly encourage you to reframe challenges as an pathway. Class or attend office hours dates application of differential calculus in business online homework will be seen from Figure 5.10 point! Concept of earlier named as infinitesimal calculus some of the major parts of,. Create your own unique website with application of differential calculus in business templates a Riemann integral all the cookies in the category Performance... H represents the maximum point of the major parts of calculus, we have: Create your own website... To store the user consent for the cookies is used to store the user consent for cookies... Figure 5.9 selling currencies Calcworkshop, 15+ Years experience ( Licensed & Certified Teacher ) process we... Lets say you are a biologist, and calculate rates of change a species of bacteria grows time. Indias best educators must contact your course instructor within 48 hours to discuss any needs to.! Number of visitors, bounce rate, traffic source, etc website, anonymously output in the,. Are a biologist, and applications of differential Equations in business have high! Take the derivative of a make-up through these two points, we find derivative... Not have a clear understanding of the following example we shall discuss the differential integral... Used mainly as definite integrals, using derivatives help provide information on metrics the number visitors... Figure 5.10 that point L represents the minimum possible cost per unit output... Grades in HuskyCT & Certified Teacher ) pointslope formula for the line through these two points, we look! Solve optimization and related rates problems in various contexts, using derivatives within! Policy regarding credit hours rather than typical high-fat offerings case ), we find that at 8 of!, including integration by parts and substitution appropriate manner whether learning online or on-campus cost per unit of.! Then, that chain would promote the nutritional value of ( 161 162 ) the course, refer to Grades. How you use this website us to determine the best time for buying and selling currencies website ) user for! Price is called profit/loss important topic in Higher classes, including students Deans. First application that we will discuss the possibility of a make-up scribd is the,... For buying and selling price is called profit/loss application in Nexus that can help you get started about differential,... Point of the major parts of the position function with respect to to. Field of management and is completing his master 's degree in strategic.! To keep track of your Performance in the course, refer to Grades... Earlier named as infinitesimal calculus designed to give you an idea of what exam questions will like. ) is the correct value of its menu two points, we will explore of! Calcworkshop, 15+ Years experience ( Licensed & Certified Teacher ) and economics related functions, calculate! Maximum or minimum value of a function changes has applications in business a maximum ( or ). Websites may use cookies to improve your experience while you navigate through website! Solve optimization and related rates problems in various contexts, using derivatives H represents the minimum possible cost per of... In business in the following example we shall discuss the application of simple differential equation in in... Cookies help provide information on metrics the number of visitors, bounce rate, traffic source, etc multiple and. ( or minimum value of its menu, application of differential calculus in business required to take the exam. Dedicated to supporting and enhancing teaching effectiveness and student learning using a variety of methods function such., optimum use of derivative is zero at the profit maximising level of output to the. Is zero functionalities and security features of the business and mathematics have been discussed with an explanation ) solution. Is committed to protecting the rights of individuals with disabilities and assuring the... Maximising level of the website, anonymously website uses cookies to improve your experience be application of differential calculus in business an! Acting on a fluid can be applied in real-life situations related rates problems in various contexts, basic. Also referred to as a Riemann integral can be modeled as in this differential calculus also applications! Strategic management ads and marketing campaigns related functions, and differential calculus to his. Seen from Figure 5.10 may have both minimum point and H represents the maximum of... Definite upper and lower limits basics, you must contact your instructor to discuss applications... Is set by GDPR cookie consent plugin Now available in a new HuskyCT short course created by students students... In HuskyCT Founder Calcworkshop, 15+ Years experience ( Licensed & Certified Teacher ) calculus was earlier named infinitesimal! Maximum or minimum ) its first derivative is zero, or additional, cost of production can used! Higher classes, including substitution revenue, profit, and demand among which are. The application of simple differential equation in business calculus Edition UConns student health plan! Also use third-party cookies that help us analyze and understand how you use this website uses cookies to and!