Graded Assignment 3: Modules 6, 7 and 8
Problem 1: For the vector valued function
r (t )  t  1, t 2 , ln(3  t ) 
a) What is the domain of the function?
b) 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 valued function.
c) 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.
d) 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) For parts (b), (c), and (d), use MVT or Maple [or Winplot or Wolfram – I don’t care for this
one] to plot the projections in both parametric and Cartesian forms. Be sure to incorporate the
“t-domain” you obtained in part (a) for the function when plotting. Use the plots to help with /
check your answers for the domain restrictions for the projections.
Problem 2: For the vector valued function
r (t )  t  1, t 2 , ln(3  t ) 
a) Find lim r (t ) .
t 2
b) Find r (t ) .
c) Compute the quantity || r(2)  r(2) || .

d) Find r (t ) dt .
Problem 3: For the vector valued function
r(t )  (t 2  6)i  e2t j  7t k
a) Find the equation of the line tangent to the curve at the value t  1.
b) Use either MVT or Maple to graph both r (t ) and your equation for the tangent on the same
set of axes, and verify that your tangent line is in fact a tangent line.
Problem 4: Solve the differential equation with initial conditions:
r(t )  1, 4t 3 ,sin t 
r(0)  2, 3,1 
r(0)  0,0, 1 
Problem 5: The vector valued function
r(t )  sin t i  te2t j  (t 3  4t )k
describes the position of an object at time t .
a) Find v (t ) .
b) Find a(t ) .
c) Find the speed of the object at t  0 .
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 and t  2 . Use MVT (numerical integration) or
Maple (symbolic) to evaluate the integral.
e) Compute the tangential and normal components of the acceleration at t  0 . Feel free to use
SciLab/Maple to do the computations.
f) Compute the curvature at t  1. Feel free to use SciLab/Maple to do the computations.
Problem 6: Suppose that the unit tangent vector for a vector valued function r (t ) is found to be
et
1
T(t )  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) .
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 , and should then convert the parametric equations to a
rectangular equation.
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) ?