math226fall07pmt2s

math226fall07pmt2s - 1 . a) Differentiating implicitly yz =...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 . a) Differentiating implicitly yz = ln( x + z ) (assuming this equation defines z as a function x and y ), find the rate of change of z in the direction of x-axis, when x = 0 ,y = 0 ,z = 1. Answer . Differentiating both sides of yz = ln( x + z ) (thinking of z as a function of x,y ) we get y z x = 1 x + z (1 + z x ) . At x = 0 ,y = 0 ,z = 1 , 1 0 + 1 (1 + z x ) = 0 or z x =- 1 . b) Show that if u ( t,x ) = f ( x + ct ) + g ( x- ct ) , then u tt = c 2 u xx . Answer . By chain rule, u t ( t,x ) = cf ( x + ct )- cg ( x- ct ) ,u x ( t,x ) = f ( x + ct ) + g ( x- ct ) , u tt ( t,x ) = c 2 f 00 ( x + ct ) + c 2 g 00 ( x + ct ) ,u xx ( t,x ) = f 00 ( x + ct ) + g 00 ( x + ct ) . 2 . The temperature at a point ( x,y ) is T ( x,y ) and T x (5 , 1) = 2 ,T y (5 , 1) = 3. A bug crawls so that its position after t seconds is given by x = 1 + t 2 ,y = cos( t ). How fast is the temperature rising on the bugs path in 2 seconds. Answer . d dt T | t =2 = T x (5 , 1) dx dt | t =2 + T y (5 , 1) dy dt | t =2 = 2(2 t ) | t =2 +3(- sin t ) | t =2 = 8 . 3 . A right circular cylinder is measured to have a radius of 10 . 02 inches and a height of 6 . 01 inches. Estimate the error in the volume calculation (the formula V = r 2 h was used) ....
View Full Document

Page1 / 4

math226fall07pmt2s - 1 . a) Differentiating implicitly yz =...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online