Mat 241 Homework Set 3 Due Professor David Schultz

Transcription

1 Mat 4 Homework Set De Professor David Schltz Directions: Show all algebraic steps neatly and concisely sing proper mathematical symbolism. When graphs and technology are to be implemented, do so appropriately. Mechanics: #.Determine the limit: lnt t π lnt t π lim,, e sin lim,lim,lim sin t t e t t t t t t t t,lim t π, esin /* L'hospital's Rle t t e,, 7 r t t, t 0 t seconds. A. On a separate graph hand sketch a trace of the vector fnction with t 0,,, and 4 seconds. #. A particle travels along the path y x

2 B. Determine the position vector and the nit tangent vector T r ( t ( t at t second and hand draw both of these vectors r t on the graph in part B. r t, 7t ; r t + 7t t r, 7 ; r r t r 7 T t T, r t r 5 5 y C. Eliminate the parameter, t, obtaining a fnction of the form yf(x and hand graph this fnction over the domain [0,4]. Is this the same graph as in part A? 7 7 r ( t t, t x t; y t x 7 x 7x so t y 8 4 y f(x(7/8*x^ x x Same crve bt the vector fnction incldes the direction of trace.

3 #. Sppose that the velocity of a particle is modeled by the vector fnction t v( t, te,sin ( 4t. Find the position vector fnction r ( t and 6 the acceleration vector fnction a ( t. t t v( t, te,sin ( 4t ; r ( t v( t dt dt i + ( te dt j + ( sin ( 4t k 6 6 looking at each component seperately: t dt dt /* long division 6 ( 6 dt dt + 6 t dt /* difference of two sqares ( 4 + ( 4 A B & ( + t ( t 4A + 4B A + B 4 A + B + B A t 4 A + B 0 B A A B A B 8 t 4 4 t 8 + dt t 4 dt 4 dt t 4 4 ln 4 + ln 4 dt t ln + c 4

4 d te dt t e e t e + c e 9 e 0 9 ( 4 ( 4 ( 4 ( 4 ( ( 4 cost 4 4 cost cos t + c 4 cos t cost + c 4 d /* cost sin t dt sin t sin t dt sin t cos t dt sin tdt sint cos tdt a t r t v ( t ( 6 cos t e t t e t cos ln,, 4t 4t + C ( 6 4t t ( 6t ( 6, te +, sin 64t, +, sin os c ( t e e 4t 4 ( t cos( t e 4 4 4

5 #4. Sppose the trajectories of two objects are modeled by: r t t, 7t, t ; r t 4t, t, 5t 6 t 0 A. Determine if these particles ever collide. r t t, 7t, t ; r t 4t, t, 5t 6 t , t t t t t t t test t in the second component. 7 they do not collide at second test t in the second component & third components They collide at seconds. B. If their paths do cross, how many times does it occr and when? Let t & s be potential times. Then the three components yield: t 4s 7t s t 5s 6 eqating& 5s 6 4s s which implies t. They only cross paths once, when t is seconds which also happens to be a collision point!

6 #5. Determine the tangent line to r ( t t t t sin ( πe,cos( πe, e t t t r ( t sin ( πe,cos( πe, e ; r ( 0 0,, ; P0 ( 0,, r ( t t t t t t πe cos ( πe, π e sin( πe, e r ( πe cos ( πe, πe sin ( πe, e π, 0, x 0 + ( π t, y + 0t, z t x π t, y, z t at t 0. Concept Development & Graphics: #6. Compte the tangent vectors and nit tangent vectors to the crves: 5 r t cos t,sin t, & s ( t cos t,sin t, 5 A. Algebraically show that these two vectors form the intersection of 4 4 the ellipsoid x + y + z and the cylinder x + y r ( t cos t,sin t, & s ( t cos t,sin t, 5 5 x cos t, y sin t, z & x cos t, y sin t, z 5 x + y cos t + sin t ; z for both x + y + z ( x + y + z they lie on the ellipsoid. The cylinder is trivial.

7 B. Use Maple or some other software package to graph the two srfaces on a grid (see helpfl code. Make sre they are both on the same graph. C. Compte the tangent vectors and nit tangent vectors to the crves. See page 858 example. Why do these two vectors have zero k components? 5 5 r ( t cos t,sin t, & s ( t cos t,sin t, r ( t sin t,cos t, 0 & s ( t sin t,cos t, 0 T sin t,cos t, 0 & T sin t,cos t, 0 s( t r t Their k components were constants. #7. I claim that T ( t T ( t. Verify this reslt for the case when r t t i t j 6 I will se some space saving variable assignment techniqes which yo shold try to tilize in yor ftre efforts.

8 r t t i + t j t, t 6 6 r t 6t t r t 9t + 6t + 4t,, ; t t 4 4 and 6 + 4t r ( t (, t t 6t t T ( t, and T ( t, 6 r ( t + 4t 6 6 Let t then t T t T t 6t, t t, ( 6 t t t 5 5 t 5 6t + t 4t t + t + 4t 48t t + t + 48t 48t ( + 4t #8. There are several space crves that are qite interesting. Use a plotting software package like Maple to plot the 4 crves listed below. A. & B. Elliptical & Conical Helices: cos ti + sin tj + tk r t t cos t, t sin t, t

9 B. & D Other spacecrves. r t sin t,( cos t,sin( cos t r t t t t t t, t sin t sin cos, sin To plot srfaces se this Maple code: Example: A. Graph the following two srfaces: The ellipsoid: and the cylinder: x + y 4 x y z > restart:with(plots:with(plottools: Warning, the name changecoords has been redefined Warning, the assigned name arrow now has a global binding > a:implicitplotd(x^/(6+y^/4+z^/4, x-5..5,y-5..5,z- 5..5,nmpoints0000,colorgray,style patchnogrid: > b:implicitplotd(x^+y^4, x-5..5,y-5..5,z- 5..5,nmpoints0000,colorsienna,stylepatchnogrid: > display(a,b,axesnormal;

10 B. Determine parametric eqations for the space crve of intersection. Then plot the srfaces and the space crve of intersection on the same graph x t y t note x + y 4 Let cos and sin : Using the above choice for x & y, find the z(t fnction(s from the ellipsoid. x y z 4cos t 4sin t z z cos t sin t z t cos t sin t and z t cos t sin t > c:spacecrve([*cos(t,*sin(t, sqrt(4-(cos(t^- 4*(sin(t^],t0..*Pi,thickness,colorblack: d:spacecrve([*cos(t,*sin(t, -sqrt(4-(cos(t^- 4*(sin(t^],t0..*Pi,thickness,colorblack: display(a,b,c,d,axesnormal;