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# Modeling of Environmental Sciences GEOG 410

UNC

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This 7 page Class Notes was uploaded by Ferne Wiza on Sunday October 25, 2015. The Class Notes belongs to GEOG 410 at University of North Carolina - Chapel Hill taught by Conghe Song in Fall. Since its upload, it has received 20 views. For similar materials see /class/228660/geog-410-university-of-north-carolina-chapel-hill in Geography at University of North Carolina - Chapel Hill.

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Date Created: 10/25/15

Derivatives and Integration 1 Derivatives Example 1 Traveling from Chapel Hill to Raleigh DistanceD30 miles leaving Chapel Hill at 800am arriving Raleigh City Hall at 845am Therefore the speed of your travel is vDt 30miles075hr40 mileshr This actually the average speed you may travel at a different speed at any particular moment If one asks what your speed is at 815am we may have to figure that out in the following way We know Dvt if we know the distance traveled from Chapel Hill at 815am D815am125miles and the distance your traveled two minute later ie D817am145miles Then the distance you traveled from 815am to 8 17am is D817amD815am2miles then the speed at 815am can be estimated as D817amD815ammiles 2 min2miles2min60 mph Which of the following is a more accurate estimate of speed you are driving at 815am a 40 mph b 60 mph Are you sure that you are driving exactly at 60 mileshr at 815am What are the situations that you can imagine that may make your speed different from 60 mph The actual speed at 8 15 may still be different from the above estimation but 60 mph is a better estimation than 40 mileshr How can we get a more accurate estimate of your driving speed at 8 15am The shorter the time you allow your car to travel after 8 15am the more accurate the speed you calculate Let t stand for time and At for the time allowed for travel the speed at 815am can be written as V N Dt At Dt t78153m At Example 2 How many of you watched a launch of a space shuttle by NASA How fast the shuttle is traveling at the time just before it is off the launch pat How fast the shuttle is traveling just before it reaches the orbit 18000 mileshour In order to get rid of the gravitation of Earth an object has to travel at an accelerating speed of 79kms2 If you do a plot of time and distance the shuttle is traveling it would look like miles seconds The Endeavour space shuttle was launched at 636pm on August 8 2007 If I ask you how fast the shuttle was traveling at 637pm how would you gure it out The shorter the At is the more accurate the speed Mathematically Dt At Dt At In general If a function yfx exists at x0 when x increased Ax at x0 ie x x0Ax the function has a corresponding increase Ayfx0Axfx0 if the limit of the ratio of Ay to Ax exists when Ax 0 the limit is called the derivative of yfx at xx0 y39 fx0 fxo fx hr lim x Axgt0 Ax Axgt0 Examples yfxC Cconstant This means regardless of what X value is y is always X Thus fXC fXAXC 1 Q1 c c0 AIxEIJAx A1301 Ax y Thus the derivative of any constant is zero yxx fXX fXAXXAX 39 Ay x Ax x Ax y 7 7 7 1 1213 Ax 1x121 Ax 1x121 Ax Hm fXX2 fXAXxAX2X22XAXAX2 y 1 Q1 x2 2xAxAx2 x2 1 2xAxAx2 2x A153 Ax A153 Ax 333 In general Xquot nXquot39 For convenience we can gure out the derivatives for the commonly used functions and put them in a table for later use so that we don t have to do this again and again Here they are C 0 X 1 Xquot nX 391 sinX c0sX c0sX sinX ex eX lnX lX The Geometric Meaning of the Derivatives y x A f 90 Ay 7 ta Ax gap When Ax O the angle p9 0L therefore the derivatives of yfx at x f c is the slope of the tangent line passing x y The functions we provided with derivative are very simple functions We often work with 2 2x more complex functions that are made from the Simple ones for example s1nx e etc we call these functions compound functions as they are functions containing functions Where sinxz can be written as sinu where ux2 Similarly e2x can be written as eu where u2x Here are the rules for taking derivatives for compound functions If yfgx is the compound function of yfu and ugx if the derivatives for ugx exists at x and yfu exists at ugx then the derivative of the compound function yfgx with respective to x is dy dy du f f gtxlt a dx du39 dx or X 11 gX Examples 1 ysinx2 and ysinu ux2 y sinu Xu sinxz XX2 2xcosxz 2 ysin2x Let usinX ysin2Xuza wan dx du dx u2 39 2u2sinx d sinx39 cosx dx z 2sin x cos x dx du dx 3 ye2xsinx Let u a du 2 dx 4 ye2XCOSZX Integration The inverse of derivatives Let me ask the inverse question in Example 1 of derivatives if I travel at 40 mph on I40 east where am I in 45 minutes how far away am I from Chapel Hill We know we traveled 30 miles in 45 min at that speed is that suf cient to know where I am What else do we need to know In derivative we can write dSdt40 The inverse of that is intergration ie j dS 40 j dt S40tC Where C is a constant determined by the initial condition eg X2 2X in fact XZC 2X where C is a constant I2xdx x2 C Similarly we can create a table of integration 1 kdx kx C n x 1 C where nil n1 3 Ie dx e C 2 jxwx 4 Icosxdx sinx C 5 Isinxdx cosx C 6 Iidx lnxC De nite Integration Given a function fX which is bounded on a b Randomly insert n points within a b so that aX0 ltX1lt ltXnb separate the interval ab into n smaller intervals X0X1 X1 X2 5 Kn1 Xn The lengths of the invervals are respectively AX1X1X0 AX2X2X1 AX XnXn1 Take any number s from any interval above calculate the product f8iAXi and sum the product S i f8 Ax Let 7 be the maximum length of the n intervals if X90 regardless of how a b is separated and how si is taken from the interval Xi1 Xi S is always approach a nite limit The limit is the de nite integration of fX on the interval a b y x NewtonLeibniz Formula 17 Ifxdx Fb Fa where fXF X 17 I f xdx is the area under the curve from a to b Examples 7r2 1 Isinxdx 0 1 2 Iaquot dx 0 3 szdx 2

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