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Calc Bus & Soc Sci II

by: Henderson Lind II

Calc Bus & Soc Sci II MATH 242

Henderson Lind II
GPA 3.8

Dev Sinha

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Dev Sinha
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This 2 page Class Notes was uploaded by Henderson Lind II on Tuesday September 8, 2015. The Class Notes belongs to MATH 242 at University of Oregon taught by Dev Sinha in Fall. Since its upload, it has received 10 views. For similar materials see /class/187177/math-242-university-of-oregon in Mathematics (M) at University of Oregon.


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Date Created: 09/08/15
MATH 2427 LECTURE 2 l ANTIDIFFERENTIATION THE INDEFINITE INTEGRAL Mathematics is full of constructions which7 once they7re understood and found useful to be done7 are also found useful to be undone For example7 subtraction undoes addition and division undoes multiplication So it should come as no surprise that it can be very useful to undo the derivative De nition 1 We say that afunction F is an antiderivative off if the derivative ofF is Example 2 The function 12 is an antiderivative of 21 The function 12 7 is also an antiderivative 2 2 of 21 The function e is an antiderivative of QIe Example 3 Name antiderivatives of 312 i and e2 Notice that in our rst two examples7 one function7 namely 21 had two antiderivatives7 namely 12 and 12 7 In fact7 any function will have many antiderivatives7 which makes taking antiderivatives different in character from taking derivatives or doing algebraic manipulations Finding antiderivatives is at rst a process of trial and error Since we know how to take derivatives7 we can often guess what an antiderivative might be7 check if our guess is correct by taking its derivative7 and then ddling around77 to get the answer just right 1 n1 Example 4 Find antiderivatives for 12 5 and e12 11 Integral notation We saw above that more than one function can be an antiderivative for a given function That may be worrisome at first7 but the following theorem puts the situation under control Theorem 5 If Fz is an antiderivative for then any other antiderivative is equal to Fz C where C is some constant function For example7 we saw that 12 is an antiderivative for 21 And so is 12 which is equal to 12777 Another antiderivative is 12 527 which is I2 7 45 It feels more natural to say that any of these antiderivatives is of the form 12 C7 where C can be any constant 0 ma e matters more or less confusing depending on your point of view7 the collection of anti derivatives of a function has another name De nition 6 The family of all antiderivatives of a function is denoted ffzdz which is also called the inde nite integral If Fz is some antiderivative of we have the equality of families of functions f Fz C We will see later why the word inde nite This notation is named as follows f is the integral sign is the integrand dz denotes the variable of integration and C is called the constant of integration Example 7 Evaluate fr 5dr and fewdr 2 FIRST APPLICATIONS OF ANTIDERIVATIVES Antiderivatives let us recover quantities from their derivatives for example total cost from marginal cost or distance travelled from velocity Example 8 The marginal cost for making Chiapets is 342 7 484 320 cents for the qth unit The setup cost for making the rst Chiapet is 2000 What is the total cost for producing ten Chiapets 1 2 MATH 242 LECTURE 2 Example 9 Basic physics dictates that neglecting air speed the downward acceleration of an object dropped due to gravity is constant 32 feet per second per second Find aformula for the distance travelled by an object dropped not thrown so it starts with no speed over t seconds How long would it take an object to fall 5000 feet for example dropping it over the edge of the Grand Canyon These examples are the beginning of the subject of di erential equations Which is the study of how to solve equations involving derivatives All of the laws of basic classical physics are Written in terms of differential equations


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