BC 1-2
Spring 2016
Problem Set #6
Name
Due: Friday, 4/22 (at beginning of class)
Please show appropriate work – no calculator allowed – except as indicated. Work should
be shown clearly, using correct mathematical notation. Please show enough work on all
problems (unless specified otherwise) so that others could follow your work and do a similar
problem without help. Collaboration is encouraged, but in the end, the work should be your
own.
 x  cosh t
1. Look at the parametric equations: 
, for t  0 .
 y  sinh t
a. Eliminate the parameter from the equations (to get an equivalent equation in terms of x
and y).
b. Sketch the graph (Neatly and accurately).
5
, then
3
find the corresponding points ( x, y ) . [Sketch the tangent line(s) on the graph in part b.]
c. Find all value(s) of t, such that the slope of the tangent line to the graph is m 
BC 1-2
Spring 2016
Problem Set #6
Name
Due: Friday, 4/22 (at beginning of class)
2. To parameterize a curve in the xy-coordinate plane means to find a set of parametric
equations whose graph is the same as the curve you are parameterizing. For example,
 x  cos  t 
 x   sin  2t 


both 
, 0  t  2 and 
, 0  t  2 are parameterizations of the


 y  sin  t 
 y  cos  2t 
unit circle x 2  y 2  1
a. Find a parameterization of the line segment from (1, 2) to (3,5) .
b. Jasmine is unwinding tape from a circular dispenser of radius a by holding the tape
taut and perpendicular to the dispenser. Parameterize the curve traced by the end of
the tape (the point P in the figure) as Jasmine unwinds the tape. You may find it
useful to use the angle  as the parameter. Assume that little enough tape is unwound
so that the radius of the dispenser remains constant.
P

BC 1-2
Spring 2016
Problem Set #6
Name
Due: Friday, 4/22 (at beginning of class)
3. Try to solve the following differential equations. That is, find all functions that satisfy the
given conditions.
2
For example: If f ( x)  2 xf ( x) . Then it could be that f ( x)  e x .
Since if f ( x)  e x , f ( x)  e x  2 x  2 xf ( x) . Note that f ( x)  Ce x works for any real
number C. This is the best answer.
2
2
a. Solve: f '( x) 
1
1  x2
b. Solve: f '( x) 
2x
1  x2
c. Solve: f ( x)  f '( x)  2 f ( x)  0
d. Solve : f ( x)  4 f ( x), given f (0)  1
2
BC 1-2
Spring 2016
Problem Set #6
4. Evaluate the following limits:
 sec2 ( x) 
a. lim 

2
x0 

x

b.
c.

lim 1  x
x 
lim
x 

3

 5
ln x
x2  1  x
 1
d. lim  1 
x 
x
x2

Name
Due: Friday, 4/22 (at beginning of class)