Download Math 2431-100 Spring 2010 – Test 3

Math 2431-100
Spring 2010 – Test 3
NAME_________________________________________________________________
NO calculators or other technology may be used on this portion of the test.
Show all work! Answers without any supporting work or explanations may not receive full credit.
If you need more space, use the back of pages 2 or 3.
1. (7 points). Find the second derivative of x  cos  t  , y  sin 2 t  .
2. (7 points). Where does x  t 2  3t  2, y  t 2 , 6  t  4 have horizontal tangent line(s) and vertical tangent
line(s)?
3. (7 points). Use the definition of cosh( x) to show that cosh 2  x  
cosh  2 x   1
2
.
  x  . Simplify as much as possible.
4. (7 points). Find the derivative of f  x   2 tan 1 sinh
5. (7 points). If two resistors of R1 and R2 ohms are connected in parallel in an electric circuit to make an R-ohm
RR
1 1 1
resistor, the value of R can be found from the equation  
or R  1 2 . If R1 is decreasing at the
R R1 R2
R1  R2
rate of 1 ohm/sec and R2 is increasing at the rate of 2 ohms/sec, at what rate is R changing when R1 = 75 ohms
and R2 = 50 ohms?
NAME_________________________________________________________________
If you need more space, use the back of pages 1 or 3.
6. (6 points).
a. Find the linearization of f  x   1  x  cos  x   3 at x = 0.
b. Use the linearization to estimate f(0.5) and f(2).
7. (7 points). Find dy: y   2  2  tanh 1   1 .

3
1
8. (6 points). Evaluate the following: cos 1  
 , csc
 2 
 2  , tan
 2 x  3 sin  x 
.
3
5  x  3 x
2
9. (7 points). Find the derivative of y 
10. (7 points). Use linearization to find the best estimate of
3
9.
1
 1 .
NAME_________________________________________________________________
If you need more space, use the back of pages 1 or 2.
11. (7 points). A storm is 50 miles offshore and its path is perpendicular to a straight shoreline. It is approaching
the shore at a rate of 4m/h. A van traveling along the shoreline wants to stay exactly 50 miles from the storm
and remain along the shoreline. The van starts at the point on the shoreline in the path of the storm. Find the
speed of the van when the storm is 40 miles from the shore.
12. (7 points). Find the equation of the line normal to y  sec( x) . At x = 4π/3.
13. (6 points). Find a cartesian equation for x  3sin  t  , y 
cos  t 
2
, 0  t  2 .
14. (6 points). Find a parameterization for the curve 3x  2  y3  6 y between the points (2, –2) and (97/3, 5).
15. (6 points). Find dr/dϑ of cos  r    .