Math 1351 011 October 15 2007 1 Announcements Homework 7 due today Homework 8 will be due Friday 10 26 2007 Exam 2 this Friday 10 19 2007 Strauss sections 2 4 through 3 7 Discussions sections this week are exam review no quiz Extra office hours this week Tuesday 10 00 to 11 30 a m Wednesday 10 00 to 11 30 a m Thursday 10 00 to 11 30 a m TTU Department of Mathematics Statistics Math 1351 011 October 15 2007 2 Related Rates Many problems involve a functional relationship y f x in which both x and y are themselves functions of another variable such as time t y t f x t We can use implicit differentiation to relate the rate of change to the rate dx dt dy dt TTU Department of Mathematics Statistics Math 1351 011 October 15 2007 3 General Procedure for Related Rates 1 Draw a figure if appropriate 2 Assign variables to the quantities that vary 3 Find a formula or equation that relates the variables 4 Differentiate the equations often implicitly with respect to time 5 Substitute specific numerical values where known 6 Solve algebraically for any required rate Examples TTU Department of Mathematics Statistics Math 1351 011 October 15 2007 4 linear approximations and differentials Recall if f x is differentiable at x a the tangent line at a point P a f a on the graph y f x has slope m f 0 a and equation y f a f 0 a x a or y f a f 0 a x a TTU Department of Mathematics Statistics Math 1351 011 October 15 2007 5 In the immediate vicinity of P the tangent line closely approximates the shape of the curve at y f x TTU Department of Mathematics Statistics Math 1351 011 October 15 2007 6 linearization If x1 is near a then f x1 is close to the point on the tangent line to y f x at x x1 That is f x1 f a f 0 a x1 a This is a linear approximation of f x at x a This process is called linearization of the function at point x a Incremental approximation formula f x1 f a f 0 a x1 a y f 0 a x TTU Department of Mathematics Statistics Math 1351 011 October 15 2007 7 Differentials We give the dy and dx of Leibniz notation quantities dy dx meaning as separate dx is called the differential of x dy is called the differential of y dx x But dy 6 y dy f 0 x dx or equivalently df f 0 x dx y is the rise of f that occurs with a change of x But dy is the rise in the tangent line relative to the change in x TTU Department of Mathematics Statistics Math 1351 011 October 15 2007 8 Differential rules Linearity d af bg a df b dg Product d f g f dg g df Quotient d fg Power d xn nx Trig d sin x cos x dx d cos x sin x dx d tan x sec2 x dx d cot x csc2 x dx d sec x sec x tan x dx d csc x csc x cot x dx Exp and log g df f dg g g2 n 1 d ex ex dx 6 0 dx d ln x 1 x dx TTU Department of Mathematics Statistics Math 1351 011 October 15 2007 9 Differentials for inverse trig d sin 1 u du 1 u2 du d tan 1 u 1 u 2 d sec 1 u u du u2 1 d cos 1 u d cot 1 u d csc 1 u du 1 u2 du 1 u2 du u u2 1 TTU Department of Mathematics Statistics
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