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PART A: MULTIPLE CHOICE: Circle the best choice. Each question is worth 2 points.
- Simplify the difference quotient
f x h f x h
, if h ≠ 0 , for f ( ) x = x^2 − 3 x + 8.
A.
f x h f x h
xh h x h h
B.
f x h f x h
x h
C.
f x h f x h
x h
D.
f x h f x h
x h
E. None of the above.
- Suppose a rectangle has its lower base on the x -axis and upper vertices on the graph of the
function y = 8 − x^2. Express the area of the rectangle in terms of x.
A. A ( x ) = 2 xy B. A ( x ) = 4 x + 2 y C. A x ( ) = 16 + 4 x − 2 x 2 D. A x ( ) = 16 x − 2 x^3 E. None of the above.
- Find an equation of a rational function f that has the following conditions:
vertical asymptotes: x = ± a x-intercept: b horizontal asymptote: y = 2 hole at x = 0
A.
( )
f x
x x b x x a
2
B.
( )
f x
x x b x x a
2
C.
( )
f x
x x b x x a
2
D.
( )
f x
x x b x a
2
E. None of the above.
- If a > 3, determine where the graph of the function
f x
x x a x
is above the x -axis.
A. (^) ( a , (^0) ) ∪ (^) ( 2 ,∞) B. (^) ( − ∞ , (^0) ) ∪( 2 , a ) C. (^) ( − ∞ , (^0) ) ∪ (^) ( 2 ,∞) D. f is always above the x -axis. E. None of the above.
- Which of the following is equivalent to f x
x x x
A. f x
x if x x if x
B. f ( ) x = x − 1
C. f x
x if x x if x
D. f ( ) x = x + 1 E. None of these.
- Which one of the following angles is coterminal with
A. 570°
B. -30°
C. 150°
D. -210°
E. None of the above.
- On a circle, an arc of length 11.52 cm subtends the central angle of 132 °. What is the area of the sector?
A. 33.0 cm^2 B. 28.8 cm^2 C. 46.4 cm^2 D. 57.6 cm^2 E. 66.0 cm^2
11. If θ = 2 3. , what is the reference angle, θ R?
A. 0.
B. 3.
C. 0.
D. 2.
E. None of the above.
- What is the range of the graph of y = − cos x +3?
A. [2, 3]
B. [2, 4]
C. [0, 1]
D. [-1, 1]
E. None of the above.
- Initially, a 25-foot ladder, leaning against a building, makes a 15° angle with the building. The ladder is moved, so that it now makes a 25° angle with the building. How far did the top of the ladder move down the building? Round to the nearest tenth of a foot.
A. 1.5 feet B. 6.3 feet C. 5.0 feet D. 1.7 feet E. None of the above.
14. sin θ
is equivalent to which of the following?
A. (^) ( )
3 sin θ −cosθ
B. (^) ( )
3 cos θ +sinθ
C. (^) ( )
3 sin θ +cosθ
D. (^) ( )
3 cos θ −sinθ
E. None of these.
- Find tan sin( −^1 x ) , x > 0.
A. x B. 1 − x^2
C.
1 − x^2
D.
x 1 − x^2 E. None of these.
- A surveyor notes that the direction from A to B is N 12° W and the direction from A to C is N 71° W. The distance from A to B is 200 meters, and the distance from B to C is 300 meters. To the nearest meter, approximate the distance from A to C.
A. 250 meters B. 349 meters C. 224 meters D. 374 meters E. None of the above.
- In tracking the relative location of two aircraft, a controller determines that the distance from the station to the first aircraft is 140 miles and the distance to the second is 110 miles. If the angle between the two aircraft is 48° 15 ′, how far apart are the two planes?
A. 95.3 mi B. 105.8 mi C. 140 mi D. 104.3 mi E. 420 mi
- A mirror for a reflecting telescope has the shape of a paraboloid of diameter 20 inches and depth of 1 inch. How far from the center of the mirror will the incoming light collect?
A. 50 inches B. 27 inches C. 25 inches D. 4 inches E. None of these.
PART C: Place your answers in the appropriate area. Each question is worth 4 points.
- Graph the function f x
x if x x if x
As x → 1 ,+ f ( ) x →_____.
23. If tan θ =
and sec θ < 0 , find the exact value of :
a) csc θ csc θ =
b) sin
^
sin
^
c) cot( − θ ) cot (^ − θ ) =
d) cos(^ θ + π) cos(^ θ + π)=
24. Solve the following trigonometric equation over [ 0 2, π ):
cos 2 θ= 3 − 7 cos θ
Answer in exact form.
25. Solve the following trigonometric equation over [ 0 2, π ):
tan 2 x − 6 tan x + 9 = 0
x ≈
Answer to 2 decimal places.
- Complete the following tables and sketch the curve.
r = 2 −2 sin θ
π 6
π 4
π 3
0
π 2 2 3
π 3 4
π
5 6
π
7 6
π 5 4
π
4 3
π 5 3
π
7 4
π
11 6
π
π
3 2
π
θ sin θ r
0 → π 2 π 2
→ π
π →
3 2
π
3 2
π → 2 π
θ r π 6 7 6
π
