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# Note for MATH 124 with Professor Long at UA

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This 11 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 19 views.

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Date Created: 02/06/15

A Chapter 6 Constructing Antiderivatives Section 61 Antiderivatives Graphically and Numerically Section 62 Constructing Antiderivatives Analytically Section 63 Differential Equations Math 124 Section 023 Fall 2007 Instructor Paul Dostert p112 A 61 Antiderivatives Graphically If the derivative of F is f we call F an antiderivative of f If f 1 then F a is an antiderivative of f 1 What about Fz1F22 Ex The graph of f is given in the below figure Sketch a graph of f in the cases when f0 O and f0 1 f x 25 15 1 p 212 A 61 Antiderivatives Numericaly We use the Fundamental Theorem of Calculus to determine integrals Recall W W in Ex Given the graph off below and that W 5 nd flt2gt f4 and f6 I0 Area 18 Area 7 Area 396 15 p 412 A 62 Indefinite Integrals Antiderivative of f x x For n y 1 since 2371 C n 12 we have mn l l I d2 0 n1 Antiderivative of f x 1x d 1 Since 11133 we have d3 1 1 dz1nlal C39 CC 1 Note We have In M not 11193 smce IS well CC defined for a lt O as well thus its integral has to be well defined p 612 A 62 Indefinite Integrals Antiderivative of f x eX d Since d e e we have CC emdaem0 Antiderivative of f x sin x and fX COS X d d Slnce SIDCL39 308 and 3082 s1112 d3 d3 we have sin2d2 CosIC39 and Cosada SiDCC C39 p 712 A 62 Indefinite Integrals Just as with derivatives sums and constant multiples can be separated in integrals we we d2 new gdiv and Cf2dzcfvda Ex Evaluate the following integrals aI45 daj b dy d 0081 sint dq p 812 A 62 Application of the FToC Now that we can evaluate indefinite integrals we can use the Fundamental Theorem of Calculus to evaluate definite integrals exactly Recall f vdv W in Ex Use the FToC to evaluate the following integrals 4 a 62 daj 0 p 912 A 63 Differential Equations A differential equation is an equation involving the derivative of an unknown function Some examples has a general solution of y2 f2 C When we are given some conditions on the differential equation to determine the specific solution then we call it an initial value problem For example if dy 2 01 div vy then we have a general solution of y2 22 0 Since y0 1 we have that C 1 p 1012 A 63 Differential Equations Ex Determine if y 6302 is a solution to the differential equation y 2mg Ex Verify that P 50 C60 20t is a general solution to P d 020P 10 dt Find the particular solution satisfying the initial condition P 60 when t 0 Ex Is there a value of n which makes y 23 a solution to the equation Ex What relationship does the following imply d y k d3 y What is the general solution to this equation p 1112 A 63 Differential Equations Ex A tomato is thrown upward from a bridge 25 m above the ground at 40 msec a Give formulas for the acceleration velocity and height of the tomato at time t b How high does the tomato go and when does it reach its highest point c How long is it in the air Ex A car starts from rest at time t O and accelerates at O6t 4 meterssec2 for 0 g t g 12 How long does it take for the car to go 100 meters p 1212

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