Calc 3 Quiz 3

# Calc 3 Quiz 3 - 5 / 2) s 2 . Furthermore, ( t ) = 1 / (5 t...

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

Instructor: Dr. Yuli B. Rudyak MAC 2313, Calculus 3 with Analytic Geometry, Fall 2011 QUIZ 3, OCTOBER 5, 2011 Problem 1. Find a vector function that represents the curve of inter- section of the cylinder x 2 + y 2 = 4 and the surface z = xy . Solution: For the cylinder we have x = 2 cos t,y = 2 sin t . Now, z = xy = 4 cos t sin t . Problem 2. Find the parametric equation for the tangent line to the curve x = t 2 - 1, y = t 2 + 1, z = t + 1. Solution: Take any point t 0 . The tangent vector at the point is h 2 t 0 , 2 t 0 , 1) i . So, the equation of the tangent line is: x = t 2 0 - 1 + 2 t 0 t , y = t 2 0 + 1 + 2 t 0 t , z = t 2 0 + t . Problem 3. Find the length of the curve r ( t ) = t 2 i + 2 t j + ln t k , 1 t e . Solution: We have r 0 ( t ) = h 2 t,t, 1 /t i . So L = Z e 1 1 + t 2 t dt = e 2 . Problem 4. Find the arc length function of the curve r ( t ) = h t 2 , sin t - t cos t, cos t + t sin t i , t [0 ,s ]. Find the curvature of the curve at t = 5. Solution: We have | r 0 ( t ) | = ( 5 / 2) t . So, L = Z s 0 = 5 tdt = (

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 5 / 2) s 2 . Furthermore, ( t ) = 1 / (5 t ). Problem 5. Find the equation of the normal plane of the curve x = t,y = t 2 ,z = t 3 at the point (1 , 1 , 1 , ). Solution: We have r (1) = h 1 , 2 , 3 i is the normal vector to the normal plane. So, the equation of the normal plane is: ( x-1)+2( y 1 )+3( z-1) = 0, or x + 2 y + 3 z = 6. Problem 6. Find the tangential and normal components of the accel-eration vector r ( t ) = t i + t 2 j + 3 t k . 1 Solution: We have a T = | r ( t ) r 00 ( t ) | | r ( t )] = 4 t 4 t 2 + 10 . Furthermore, a N = | r ( t ) r 00 ( t ) | | r ( t )] = 2 10 4 t 2 + 10 ....
View Full Document

## This note was uploaded on 10/20/2011 for the course MAC 2313 taught by Professor Keeran during the Fall '08 term at University of Florida.

### Page1 / 2

Calc 3 Quiz 3 - 5 / 2) s 2 . Furthermore, ( t ) = 1 / (5 t...

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

View Full Document
Ask a homework question - tutors are online