Lecture 1 - MATH 118, LECTURE 1: Review &...

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Unformatted text preview: MATH 118, LECTURE 1: Review & Integration by Substitution 1 Course Information For a detailed breakdown of the course content and available resources, see the Course Syllabus (general course folder). Other relevant information for this section of MATH 118 is: Instructor : Matthew D. Johnston Office : MC 5126 Office hours : 11:30-1:30 M (negotiable) Tutorial : 12:30-2:30 Tu (RCH 305) (NOTE: the first tutorial will be Jan. 13) 2 Review We will be studying integration for the first 3-4 weeks of this course. Topics which should already be familiar to you from MATH 116 include: Indefinite integral (anti-differentiation): integraldisplay f prime ( x ) dx = f ( x ) + C. Definite integral (area under the curve): integraldisplay b a f ( x ) dx = lim bardbl x i bardbl n summationdisplay i =1 f ( x * i ) x i , Fundamental Theorems of Calculus: FTC #1: integraldisplay b a f ( x ) dx = F ( b )- F ( a ) where F prime ( x ) = f ( x ) . FTC #2: d dx integraldisplay x a f ( t ) dt = f ( x ) . 1 You should also be familiar with the various integral laws found on pg. 390 of Trim. So we know integration undoes differentiation (FTC #1) and differ- entiation undoes integration (FTC #2). Furthermore, the Fundamental Theorem gives us a geometrical interpretation of the (somewhat abstract) process of anti-differentiation. We can find the area under a curve by evalu- ating the anti-derivative of a function at the end points and subtracting (at least for continuous functions). Throughout this course, we will be see many applications of both of these interpretations of the integral (anti-derivative and area under the curve). Knowing the relationship between derivatives and integrals, however, does not get us any closer to understanding how to integrate. We might first hope that we can develop methods of integration in the same way we developed them for differentiation. With differentiation, we were able to derive formulas (e.g. chain rule, product rule, quotient rule, etc.) which allowed us to differentiate complicated functions so long as we knew how to differentiate the individual components. For example, consider the function f ( x ) = sin ( x 2 ) e- x ....
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Lecture 1 - MATH 118, LECTURE 1: Review &...

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