1. Find the limit. a.
normal and the rectifying planes at the point that
lim , →
corresponds to that value of .
a. = cos + sin + , =
, sin 2
→
b. lim ,
b. = + cos + sin ,
2. Find the first and second derivatives.
=0
a. = + tan +
11. Find Κ at the specified point.
b. = tan + cos − $
a. = 3 cos + 4 sin + ; =
3. Find parametric equations of the line tangent to
b. = + + , = 0
the graph of at the point where = % .
c. = 2 sin 2 + 3 sin , =
a. = ln + + ; % = 2
13. At what point does F = G have maximum
4. Find a vector equation of the line tangent to the
curvature?
graph of at the point (% on the curve.
14. Find the velocity and acceleration vectors and the
a. = 2 − 1 + $3 + 4 ;
b. = − (% 4,1,0
5. Given = + + . Ă— .. 0.
function is the position vector. a. = 3 cos + 3 sin , =
6. Given = 2 + 3 + 2
, =
1
,
find 2 3 â‹… 4 and 2 3 Ă— 4.
7. Evaluate the following integrals. 5.77 8 9 − 2 +
<.77 8 9 3 â&#x2C6;&#x2019; %
3 +
+
a.
B
15. Find the position and velocity vectors given the
; 777K 0 =
b. H = + ; 77J 0 = 2 + ; K 0 =
: ;
â&#x2C6;&#x2019;
c. H = + 1 â&#x2C6;&#x2019; ; 77J 0 = 3 â&#x2C6;&#x2019;
= 12 â&#x2C6;&#x2019; 2 , â&#x20AC;˛ 0 = 2 â&#x2C6;&#x2019; 4 ,
0 = >
; 77K 0 = 2
16. Find the scalar and vector tangential and normal components of acceleration at the stated time.
9. Find ? and @ at the given point.
=
a. H = â&#x2C6;&#x2019; cos â&#x2C6;&#x2019; sin ; 7J 0 =
8. Solve completely for . ..
d. = sin + cos + ,
ff acceleration vectors and initial conditions.
1 : ;
+
c. = + + , = 1
â&#x2020;&#x2019;
B
b. = 2 + 4 + 1 â&#x2C6;&#x2019; , = 1
, find lim, â&#x2039;&#x2026;
2
speed at the indicated time . The given vector
+ 4 â&#x2C6;&#x2019; ;
B
sin 3E, at E = 0.
(% â&#x2C6;&#x2019;1,2
B
12. Find the curvature of the polar equation D =
b. = 2 cos & + 2 sin & + 3 ; % =
B 1
a. = cos + sin +
Find as well the curvature of the path at the point
,
= 0.
b. = ln + , =
10. Find ? , @ , A for the given value of . Then find the equations for the osculating,
where the particle is located at the stated time t. a. = + ; 7 = 0 b. = cos + sin ; 77 =
c. = + + ; = 1
$B
d. = 3 sin + 2 cos − sin 2 ; 77 = B
e. L = −4 , H = 2 + 3 f.
L = 2 + 2 + , H = + 2
17. A shell is fired from ground level with a muzzle speed of 320ft/s and elevation angle of 60º. Find the parametric equations for the shell’s trajectory, the maximum height reached by the shell, the horizontal distance traveled by the shell, the speed of the shell at impact. 18. A rock is thrown downward from the top of a building, 168 ft high, at an angle of 60 º with the horizontal. How far from the base of the building will the rock land if its initial speed is 80ft/s? 19. A shell is to be fired from ground level at an elevation angle of 30 º. What should the muzzle speed be in order for the maximum height of the shell to be 2500 ft.? 20. Find two angles of elevation that will enable a shell fired from ground level with a muzzle speed of 800ft/s to hit a ground-level target 10000 ft away. 21. A ball rolls off a table 4ft high while moving at a constant speed of 5ft/s. a. How long does it take for the ball to hit the floor after it leaves the table? b. At what speed does the ball hit the floor? c. If a ball were dropped from rest at table height just as the rolling ball leaves the table, which ball would hit the ground first? -EAArances