DOC PREVIEW
GT MATH 2401 - A very gradifying lecture
School name Georgia Tech
Pages 29

This preview shows page 1-2-3-27-28-29 out of 29 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 29 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 29 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 29 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 29 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 29 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 29 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 29 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Lecture 11 MATH 2401 - Harrell Copyright 2008 by Evans M. Harrell II. A very gradifying lectureScenes from our previous episode  Tangent and normal vectors,  Tangent and normal linesTips 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. Like…..intersecting surfaces.  The intersection of two surfaces is usually a curve. How is it oriented – i.e., what is its tangent direction? Monday was fun, so let’s do some new (even more fiendishly intricate) geometric problemsIntersection 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(π y z) and z = 2 x2 – 4 y2 + ¼ as it passes through (1, ¼, 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 If f is differentiable on the line segment ab, there exists a position c so that f(b) - f(a) = ∇f(c)•(b - a)  Why is this not as great as in 1D? A: the gradient doesn’t have to point in the direction (b - a), and |∇f(c)| isn’t the average slope |f(b) - f(a)|/|b - a|. Mean value theorem If f is differentiable on the line segment ab, there exists a position c so that f(b) - f(a) = ∇f(c)•(b - a)  Example: (0,0) to (1,0), f(x,y) = x+y. Mean value theoremGradient determines f up to a constant  Let U be open and connected, and f and g be differentiable on U. If ∇f = ∇g on U, then f(x) = g(x) + C.  How would you prove this?  Good enough to replace f by f-g and show that is ∇f = 0 on U, then f is a constant.  Connect any two points by “polygonal path” and use M.V. Theorem.Which = 0.Gradient determines f up to a constant  Let U be open and connected, and f and g be differentiable on U. If ∇f = ∇f on U, then f(x) = g(x) + C.  How about an example?  Arctan(y/x) vs.


View Full Document

GT MATH 2401 - A very gradifying lecture

Download A very gradifying lecture
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view A very gradifying lecture and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view A very gradifying lecture 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?