Download Test 6 2008

MIFE 129
Test 6
12/10/08
Name__________________ PART I _____/70
This part must be done without a calculator. Show all work.
1. Given sin u 
4
3
,  u
, find the exact values of:
5
2
i)
sin 2u
iv)
sin
u
2
ii)
(2 pts each)
cos 2u
cos
v)
iii)
tan 2u
u
2
vi) Which quadrant is the angle 2u in?
Which quadrant is the angle u/2 in?
2. Find all solutions to sin 2x sin x  cos x on 0  x  2 .
(6 pts)
3. Suppose x is in the second quadrant and y is in the third quadrant, cos x = -4/5 and csc y = -3.
i) Find the exact value of sin(x+y).
ii) Find the exact value of cos(x+y).
iii) In what quadrant is x+y?
(6 pts)
4. Find the limits.
(6 pts)
 sin( 3x) 

x 0
 5x 
i) lim 
 1  tan x 

x  sin x  cos x 
4
ii) lim 

5. Find dy/dx.
i) y 
(4 pts each)
 
cos 5x3
x
iii) y  x2 arcsin  3x 
ii) y 
1 2
sin  2 x 
2
iv) y = tan(xy)
6. Find the equation of the line tangent to f(x) = esin x at (0,1).
(6 pts)
7. Evaluate the integrals.
(4 pts each)

i)
  3t  5sin t  dt
4
ii)
 3sec
2
 d
0
8. i) Sketch the graph of y  2cos  x  for one full period starting at x=0.
ii) Shade the area bounded by the graph of the function and the x-axis on 0  x  3/ 2 .
0.5
iii) Use the Fundamental Theorem of Calculus to evaluate
 2cos  x  dx .
0
iv) Explain how you can use the value of the integral in (iii) to calculate the shaded area in (ii).
v) Write the shaded area in terms of integrals.
(2 pts each)
Math 129
Test 6
12/10/08
Name____________________
PART II ______/30
You may use a calculator on this part. If the calculator was used, write the expression that you keyed in.
9. An engineer wants to measure the width AB of a sinkhole. Suppose that he places a stake at point C and
determines the measurements shown in the figure. How wide is the sinkhole?
(5 pts)
C
103o
80 ft.
110 ft.
A
B
10. Solve the triangle. If there is no solution, clearly explain why. If there is more than one solution, give all
solutions. Note: the triangle is not necessarily a right triangle.
(5 pts each)
i. a = 11, b = 14, c = 20
C
b
a
A
B
c
ii. A = 20, a = 12, and b=30
11. Prevailing winds affect the growth of a tree. An observer, due south of the tree, notes that the tree is
leaning east at an angle of 94 degrees. From a point 35 meters west of the tree, the angle of elevation to
the top of the tree is 23 degrees. What is the height of the tree?
(5 pts)
12. An airplane flies at an altitude of 5 miles toward a point directly over an observer. θ is the angle of
elevation from the observer to the airplane; x is the ground distance from the observer to a point directly
under the plane.
(5 pts)
i) Write θ as a function of x.
ii) The speed of the plane 400 mph. Find dθ/dt when x is 10 miles. (Show the units.)

13. Verify the identity sin  4 x   4sin x cos x 1  2sin 2 x

(5 pts)