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# CALC ANALYT GEOM II MATH 125

UW

GPA 3.76

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This 6 page Study Guide was uploaded by Addison Beer on Wednesday September 9, 2015. The Study Guide belongs to MATH 125 at University of Washington taught by Matthew Conroy in Fall. Since its upload, it has received 20 views. For similar materials see /class/192067/math-125-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Summary for Midterm One Math 125 Here are some thoughts I was having while considering what to put on the first midterm The core of your studying should be the assigned homework problems make sure you really understand those well before moving on to other things like old midterms o 42 Mean Value Theorem You should understand what the Mean Value Theorem says and be able to give an example of at least one application of it o 410 Antiderivatives You should know what it means for f to be an antiderivative of Given two functions f and gz you should be able to say whether or not f is an antiderivative of How many antiderivative does a function have What is that quot0quot business all about 0 51 Areas and Distances How can we approximate the area of a region in the plane What is an interpretation of the area under the graph of a velocity function 0 52 The Definite Integral You should understand the definition of the de nite integral and its relation to area under a curve You should be able to use the midpoint rule to approximate a definite integral Problems 35 40 are particularly nice o 53 The Fundamental Theorem of Calculus Part 1 If f is continuous on 17 then gltzgt j N dt is continuous on 17 and differtiable on 17 b g f Part 2 If f is continuous on 17 then where F is any antiderivative of f Be sure you can differentiate functions like g 6 2 dt using the chain rule and part 1 of the FTOC see eg problems 50 52 0 54 Indefinite Integrals and the Net Change Theorem Here we get the notation that fltzgt dz stands for the most general antiderivate of f o 55 The Substitution Rule The substitution rule is the most important and powerful tool for finding antideriva tives It can be considered to a certain extent the reverse of the chain rule for differ entiation Substitution is a way of getting from one indefinite integral to another When trying to find antiderivatives we may need to try several different substitutions until hit ting on one that improves the integral we are working with to the point that we can find the antiderivative Sometimes more than one substitution used in sequence is an effective way to go Practice will improve your ability to see the right substitu tions As we get more techniques for finding antiderivatives the substitution method will always be with us It will pay to make sure you can use the method well now There are tons of practice problems in this section to work on to improve your sub stitution ability eg problems 7 44 and 49 70 o 61 Areas between Curves The first of our many applications of the integral is to find the area between curves quotArea is the integral of width In many instances you will want to express the area as an integral in y rather than in z It very often helps to have a decent sketch of the region whose are you are trying to find Note that b b ltfltzgt 7 gltzgtgt dz e dz 7 fltzgtgt dz so if your answer comes out negative which is impossible for an area check that you haven39t got the difference of the two functions in the wrong order o 62 Volumes by Cross Section Here is developed the idea that quotvolume is the integral of cross sectional area Although many of the examples we looks at involve solids of revolution whose cross sections are circles this method applies to any solid that has cross sections whose area can be expressed as a function of z or y o 63 Volume by Cylindrical Shells The method of washers disks is great but in certain cases we can result in an inte gral we are unable to evaluate or indeed to setup So we have another method the method of cylindrical shells Even if the washer disk method works the cylindrical shells method can be easier Practice will help you decide which method to use on a given solid Problems 7 14 on page 447 are good for this Summary for Midterm Two Math 125 Autumn 2006 Here is an outline of the material for the second midterm The core of your studying should be the assigned homework problems make sure you really understand those well before moving on to other things like old midterms o 64 Work Many problems finding the work required to perform a certain task can be solved by cutting the task into pieces approximating the work to perform each piece summing these approximations and taking a limit as the number of pieces goes to infinity The result is of course an integral There are basically three categories of problems you might see in this course cable problems spring problems and pumping digging problems We covered pump ing digging problem for midterm 1 the others might appear on midterm 2 and the final 0 65 Average Value of a Function This is a very short section You should understand the definition of the average value of a function on an interval 0 71 Integration by Parts You should understand how to apply the integration by parts technique udvuvivdu You should be able to recognize integrands for which this technique is particularly appropriate such as 1 a positive integer power of z times sin z cos x e or related functions 2 lnz times any power of z 3 5 times sin z or cos x Integration by parts can be the method of last resort if nothing else works you can almost always try quotpartsquot if you can differentiate the integrand you can use it Whether it helps or not is another question but it works well with such function as In x and arcsin x o 72 Trigonometric Integrals You should be able to integrate integrals of the form sinquot z 003 quot x dx and tanquot z seem x dx The strategies for these integrals depend on the purities of m and n that is whether they are even or odd You should practice all cases Note that the case where the power of tan z is even and the power of sec z is odd does not have a clear strategy so a variety of techniques may be needed 0 73 Trigonometric Substitution This technique exploits trigonometric Pythagorean identities to give useful substitu tions that convert quadratic expressions into squares of trig functions This makes it particularly useful for eliminating square roots ie when you see a quadratic ex pression inside a square root there39s a good chance that a trig substitution might be useful One thing that makes this technique more work that a quotsimplequot substitution is the work required to convert back to the original variable once an antiderivative is found One efficient way to do this is the quottriangle methodquot as described in lecture and the text Be sure to practice this aspect of this technique 0 74 Partial Fractions This technique is applicable to integrands that are rational functions You should know how to apply this technique to any rational function with a de nominator that is factorable as a product of linear factors The idea behind this method is completely algebraic a rational function whose de nominator is a product of linear factors can be expressed as the sum of simpler ra tional functions each of which has a denominator which is linear and a constant numerator Such simpler functions are easily integrated Long division of polynomials is a necessary first step when the numerator has de gree equal to or greater than the degree of the denominator o 75 Strategies for Integration This section is perhaps the most important On the exam the integrals will simply be presented to you for you to solve You will have to decide which technique or techniques to apply to find the antiderivative There are 81 problems in this section and almost all are good practice You can skip problems 10 26 61 and 71 since these involve cases of the partial fractions method that we39re skipping o 77 Numerical Integration When we want to evaluate a definite integral and we cannot find an antiderivative of the integrand we can approximate the value of the integral Three techniques we39ve seen for doing this are gtk Midpoint Rule gtk Trapezoid Rule gtk Simpson39s Rule All three methods are quite similar and require us to quotsamplequot the function at a number of equally spaces values of x then combine these value according to a cer tain formula which depends on the method 0 78 Improper Integrals There are two types of improper integrals 1 Those of the form from L Wfltzgtdzorfltzgtdz 2 Those of the form b a f dx where f is discontinuous somewhere on the interval 11 We treat improper integrals by defining them as the limits of quotproperquot integrals For instance 0 fx dx dx

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