Graded Assignment 3: Modules 6, 7 and 8
Again, please answer on your own paper (neatly and well organized), OR, make yourself a custom copy
of the Word doc with more spacing and fill in. Do not cram things onto this page, and do not put half
your assignment on this page and the other half on some other pages – this all needs to be in order and
easy to read. Thanks!
Problem 1: For the vector valued function
r (t )  x(t ), y (t ), z (t )  3  t , t 3 ,ln(t  2) 
a) Write each of the component functions, and first give the natural domain for each, using correct
interval notation. Please do NOT write on this page (unless you are modifying the whole document
to accommodate ALL your answers) – I’m using the setup below to indicate what I’m looking for and
how you should write the answer, but write it on your own paper.
x(t ) 
Domain of x ( t ) is t 
y (t ) 
Domain of y ( t ) is t 
z (t ) 
Domain of z (t ) is t 
b) What is the overall domain of the function r (t ) ?
c) This is the tricky bit for people – whatever that domain is, that now becomes the restricted domain
for EACH of the component functions individually. Using that restricted domain, work out what the
range of each component function is on that domain. You may find it helpful to make quick 2D plots
(in whatever software you like) of each of the component functions to accurately determine the
range, but you don’t need to include them. Please clearly indicate in your answer:
Range of x ( t ) is x
Range of y ( t ) is y 
Range of z (t ) is z 
d) The reason we need to those ranges is that, when we do projections and convert the equations to
Cartesian form, the ranges are the restrictions on x , y , and z . Note the important point there –
we’re not trying to figure out the restrictions from the parametric equations (or the Cartesian
equation), we already know the restrictions, and are applying them TO those equations.
i.
Write the parametric equations that describe the projection of the curve onto the xy plane,
and write the corresponding Cartesian equation (an equation in x and y ). Be sure to
indicate any restrictions on the values of x and y that come from the original vector
ii.
iii.
valued function.
Write the parametric equations that describe the projection of the curve onto the xz plane,
and write the corresponding Cartesian equation (an equation in x and z ). Be sure to
indicate any restrictions on the values of x and z that come from the original vector
valued function.
Write the parametric equations that describe the projection of the curve onto the yz plane,
and write the corresponding Cartesian equation (an equation in y and z ). Be sure to
indicate any restrictions on the values of y and z that come from the original vector
valued function.
e) Use MVT to plot the vector valued function r (t ) on the domain you determined, and then use the
built in projection buttons to flatten it and get the 2D graphs of the xy , xz and yz projections. Do
screenshot and attach all of these plots (four plots total from 3d + projections). Make sure
everything makes sense in your previous answers – you should see the x , y , and z restrictions
appearing naturally as part of the flattened plots.
Problem 2: For the vector valued function
r (t )  e2 t i 
2
2
t
t
j  t 2 ln(7t  1) k  e2t ,
, t 2 ln(7t  1) 
t 3
t 3
a) Differentiate to get r '(t ) . You should definitely check each piece with Maple/Wolfram (no
excuse for having the wrong expression), but please show the differentiation process by hand
(products, quotients, chains).
b) Evaluate r (1) and r '(1) (approximate values to two decimal places)
Find the equation of the line tangent to the curve at the value t  1. Write answer in
parametric form.
d) Use either Maple (preferred) or MVT (if you don’t have access to Maple) to graph both r (t ) and
c)
your equation for the tangent on the same set of axes, and verify that your tangent line is in fact
a tangent line. Be sure to attach the plot.
Problem 3: The vector valued function
v(t )  sin t i  tet j  (3t 2  4)k  sin t , tet , 3t 2  4 
describes the velocity of a moving object at time t (assume units for velocity are m/s).
a) Given also the initial position of r (0)  0,0,5  , find the function for the position of the object
r (t ) . Note that that’s an initial value problem, and also, you’ll need an integration by parts on
the middle component.
b) Find the acceleration function a(t ) .
c) Find the speed of the object at t  1 s (decimal approximation to two places).
d) Set up (do not evaluate by hand) the integral that gives the distance traveled by the object (aka
the length of the curve) between t  0 s and t  2 s. Use MVT (numerical integration) or
Maple (symbolic) (or Wolfram, or your calculator – whatever works) to evaluate the integral.
e) Evaluate v(0) and a(0) , and use them to compute the tangential and normal components of
the acceleration ( aT and aN ) at t  0 .
f)
Compute the curvature at t  1 s. Note that I have changed the t value and you can’t reuse the
previous quantities. These will not be “nice” numbers, and you should set it all up, but use
SciLab or Maple to do the computations.
Problem 4: Suppose that the unit tangent vector for a vector valued function r (t ) is found to be
T(t ) 
et
1
, 2t

2t
1/2
(e  1) (e  1)1/2
with
T(t ) 
et
e2t
,

(e2t  1)3/2 (e2t  1)3/2
a) Give expressions for v (t ) and || v (t ) || There is actually more than one possible answer to this there is, however, one fairly obvious answer - the subsequent parts should follow from your
answer here.
b) Find N (0) . Notice that all the Calculus you need (differentiation) is done. It is OK (and
preferred) to start plugging in values before you perform vector operations - makes for much
less algebra.
c) Find B (0) .
d) Sketch a graph of r (t ) that passes through the point r (0)  0, 2  . To do this, you’ll need to
first get the vector valued form of r (initial value problem, starting from v ( t ) ), and should then
convert the parametric equations to a rectangular equation. Please show that – I want to see
the Cartesian (rectangular) equation for the graph you are graphing.
e) Sketch the vectors T(0) , T(0) , and N (0) on your graph (tails of all three should be at r (0) ).
f)
What is the direction (in or out) of B (0) ?