GT MATH 2401 - A very gradifying lecture (29 pages)

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A very gradifying lecture



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A very gradifying lecture

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Lecture Notes


Pages:
29
School:
Georgia Institute of Technology
Course:
Math 2401 - Calculus III

Unformatted text preview:

MATH 2401 Harrell A very gradifying lecture Lecture 11 Copyright 2008 by Evans M Harrell II Scenes from our previous episode Tangent and normal vectors Tangent and normal lines Tips for the test don t lose points for trivial reasons Show some work There will be no partial credit for what isn t shown Make sure the grader sees the key facts formulae somewhere Check your algebra calculations Common sense An answer has to be the right kind of animal Put the answer where indicated Monday was fun so let s do some new even more fiendishly intricate geometric problems Like intersecting surfaces The intersection of two surfaces is a curve How is it oriented i e what is its tangent direction usually Intersection of a plane and a cone Tangent to the curve at the intersection of 2 surfaces such as a plane and a cone the classic conic sections But what about The tangent to the curve where x sin y z and z 2 x2 4 y2 as it passes through 1 2 The chain rule s d dt f r t f r t r t Just like 1 D In components df dt f x dx dt f y dy dt f z dz dt Examples The chain rule s Suppose the temperature in a plate is T x y 4 x2 2 x y 4 y2 and that an object moves in a circle r t 2 cos t i 2 sin t j At what rate is the temperature changing The chain rule s What about u x y where x and y depend on s and t For example a change of variables The chain rule s What about u x y where x and y depend on s and t u s u x x s u y y s Remember Add up all the possible routes for connecting u to the independent variable Word problems Suppose that price of your widgets is P t you are selling at a rate of R t per month and your expenses are F t c t R t How rapidly is your profit changing if P 2 R 3000 F 2500 c 1 P t 1 R t 20 c t 05 and F t 5 Word problems Suppose that price of your widgets is P t you are selling at a rate of R t per month and your expenses are F t c t R t Profit PR F c R d Profit dt R P P c R F c R Mean value theorem If f is differentiable on the line segment ab there exists a position c so that



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