Applicationsof Differentiation and Integration in Daily Life
Theword ‘Calculus’ is not a common word for everyday use. Moststudents have a difficulty in understanding the concepts of calculussince they have difficulties for imagining the real world applicationof calculus. Most of the people denote that calculus is only for themathematicians and the physicists. We heard the word differentiationand integration mostly in studies but in fact, it can be seeneverywhere from its uses. The real life application method ofteaching calculus has been a successful method for efficientunderstanding for the students.
Calculusis the basic study of the rate of change. Differentiation andintegration is the two branches of calculus. Differential calculus isconcerned for the rate of change and finding the slopes of the curveswhile integral calculus or inferential calculus is concerned in theaccumulation of quantities and finding the area under the curve.Isaac Newton and Gottfried Leibniz are the developers of the moderncalculus. The fundamental theorem of calculus converges the relationof differential and inferential calculus. It states the two branchesare inverse operations. (Larson and Bruce, 2010)
Thecomputation of differential equations includes finding the velocity,acceleration, slope, maximum amount and minimum amount. Thecomputation of integral equations involves area of the curve, work,pressure, and center of mass. Advanced calculus may also be involvedin the power series and Fourier’s series computation. Calculus isused in different sciences whenever a mathematically modeled equationappears and it needs an optimal solution. (Larson and Bruce, 2010)
Themain application of the concepts of calculus is used in studying mostof the engineering concepts. Engineering concepts involvesdifferential equations with different orders to evaluate and analyzethe basic concepts such as thermodynamics, strength of materials andfluid dynamics. However, calculus is one of the most used concepts ineveryday life. We cannot see the concept in a literal or problemsolving way but we can observe it in their basic uses andapplications. (Larson and Bruce, 2010)
Application of Calculus
Findingthe slope of the curve (Differentiation)
Findingthe slope of a line is very basic and can use algebra to solve it.However, the slope of the curve cannot be solved using basic algebra.Calculus has a standard method of solving the slope of a curve. Itallows us to know how steep the curve was at an instantaneous time orpoint. Most of the graphs created in a real world consist of curvesand the line graphs are rare. (Tan,2007)
Theconcept of the slope of the curve is involved in the calculations ofdifferentiation. Differentiation can be used to find the relationshipbetween two or more variables. In the linear algebra equation of they = mx + b, the slope m can be defined as
Thechange we can observed is only limited and be easily seen when thechange occurs for a very long time. For example, the change of heightcannot be observed from day to day. But if you observed your height 5years ago and your height at present, you can observe the change.Calculus finds a way of observing the change or measuring the amountor quantity at a given time. They created an infinitesimal intervalwhere you can compute the amount for any given instant. Thedefinition of the slope of the line can be defined as:
Whenwe are trying to observe a very small change in x, say zero, thefunction becomes:
Thefunction of the derivative can be used to find the slope of thecurve. Finding the slope of the curve is important in observing thechanges at an instant period of time. The application of theseconcepts can be seen in the road signs. (Tan, 2007)
Figure1 Road sign application of slope of the curves.
Theslope of the roads is computed using analysis in calculus. Thepercent steepness is used by road safety managements to categorizethe curves and steepness of the roads. Road signs are just an exampleof where calculus can be useful. Finding the slope of the curve canalso be important in predicting the amounts or value of certainvariables at any given instant. The application of derivative is verywide in the aspects of transportation and travels. This is the reasonwhy the calculus is thought with speeds and velocities. Theprediction concept of the amount at instant can be used to predictthe speed and velocity at a given period of time. This is the conceptbehind the speedometer.
Findingthe Area Under the curve (Integration)
Findingan area of a definite shape is an easy task for everyone. But findingthe area of an indefinite curve is an almost impossible task. It wascalculus that created a method for solving the area of a curve. Thereare two type of integral function, the definite and indefiniteintegral. Definite integral is used to compute for the area createdby the curve and its boundaries. (Tan, 2007) The symbol used forcomputation of integral function is which denotes sum and the definite integral can be represented as:
Thebounds of the curve is denoted by the values a and b. otherassumptions should be consider before the calculation of the integralfunction such as the curve must be continuous between points a and b.the value solve by the equation is the area under the curve:
Thevalue of the function F(b) and F(a) are the derivative values of thefunction at values b and a, respectively. It can be represented in agraph. (Tan, 2007)
Figure2. Graphical representation of the area under the curve
Thevalue of S denotes the area under the curve. The concept of integralcan be used to find the values of curves and function representingstatistics or observed value. For example, the biologist tends tofind the total growth of bacteria from the growth curve created bythe values or using computers to create the curve. The curve createdin engineering can also use calculus to find the center of mass of anobject. The center of mass is very essential in engineering designand material construction. Integral function can also be used infinding the volume of a given solid by computing the integralfunction of its area. (Tan, 2007)
Optimizationis one of the most basic applications of calculus. The basic conceptof optimization is finding the maximum and minimum values of afunction. It is a systematic approach for mathematical computation ofthe best fitted value for a wide range of functions which is thebasic aspect of applied calculus. Example, in finding the best heightand width of a rectangular material to find the maximum volume thatcan be occupied, calculus is used to solve such problems. (Tan, 2007)
Figure3. Illustration of an optimization problem
Inthis problem, the optimize value is solve using the concept ofderivative. In the derivative function, there is a critical value inwhich the value of its derivative is zero. This critical value mayimply a change from increasing order to decreasing order or viceversa. These critical points can be analyzed if they are the optimumvalues. The test for the critical values is the first derivativetest. (Ernest, 2002)
Inengineering, optimization is quite useful for volume and pressureproblems. In designing materials and buildings, optimal measurementsis used to find the maximum amount of force and pressure that canwithstands by the created material or support of a building. Theseproblems refer to as rigid body problem. Another application inengineering is for the computation of the maximum power and work thatcan be collected in the fluid thermodynamics of refrigeration andwork production using steam as an energy source. (Ernest, 2002)
Ineconomics, the problem of optimization appears in creating a perfectcombination of business process to produce the maximum profit for thecompany or organization. For example, Calculus can compute theminimum amount of materials to be used in creating the maximum amountof products to be produced. Thus it does not only compute for theminimal cost of materials but to maximize the production to createmore profit. Calculus is also used to compute marginal cost andmarginal value problems on economics. (Tan, 2007)
Calculusis a basic language used in mathematics, engineering and economics.However, people may not observe that calculus is a function which hasa wide variety of uses in our daily lives. In credit card banking,parameters such as fluctuating balances and changing interest ratescreates a problem in optimization. The credit card companies usescalculus to create a program that can compute the minimum amount ofpayment at the exact given time. The parameter use in laboratoryscale such as bacterial growth rate, temperature and food conditionare problems that can be solved using differential calculus. Thebiologist may increase the bacterial growth rate by changing theconditions using optimization equations or the biologist may decreasethe growth rate for eliminating its harm and potential risk. Tocreate an exact positioning of the cable wires which is miles apart,the electrical engineers use integral equations. Cables usually hungfrom its poles and create curves which results to difficulties incomputation of exact length to reduce the material use and maximizethe length to be connected without the use of calculus. in structuresand buildings, engineers uses optimization problems to find theminimum amount of materials necessary to create the building and alsomaximizing the weight and pressure that the building can endure. Themanufacturers also determine the center of mass of the materials andproducts they are producing. In other words, most of the products weare using right now are in a way a work of calculus computations.
Applicationof Calculus, University of Nipissing. Retrieved May 6, 2014 fromhttp://calculus.nipissingu.ca/calc_app.html
ErnestP. (2002). Empowerment in mathematics education, Philosophy ofMathematics Education Journal.
Larson,R., Bruce E. (2010). Calculus,9th ed., Brooks Cole Cengage Learning. ISBN978-0-547-16702-2
TanS. (2007). Applied Calculus for the Managerial Life, and SocialSciences, 7 ISBN-13: 978-0-495-01582-6
Weisstein,E. Slope,MathWorld. Retrieved May 6, 2014 fromhttp://mathworld.wolfram.com/Slope.html