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Unformatted text preview: Math 125 Summary 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). • 4.10 - Antiderivatives – You should know what it means for f ( x ) to be an antiderivative of g ( x ) . – Given two functions, f ( x ) and g ( x ) , you should be able to say whether or not f ( x ) is an antiderivative of g ( x ) . – How many antiderivative does a function have? – What is that ” + C ” business all about? • 5.1 - 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? • 5.2 - The Definite Integral – You should understand the definition of the definite 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. • 5.3 - The Fundamental Theorem of Calculus – Part 1: If f is continuous on [ a, b ] , then g ( x ) = integraldisplay x a f ( t ) dt is continuous on [ a, b ] and differtiable on ( a, b ) , g ′ ( x ) = f ( x ) . – Part 2: If f is continuous on [ a, b ] , then integraldisplay b a f ( x ) dx = F ( b )- F ( a ) where F is any antiderivative of f . – Be sure you can differentiate functions like g ( x ) = integraldisplay x 3 sin x e t 2 dt using the chain rule and part 1 of the FTOC (see, e.g., problems 50-52). • 5.4 - Indefinite Integrals and the Net Change Theorem – Here we get the notation that integraldisplay f ( x ) dx stands for the most general antiderivate of f . • 5.5 - 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....
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