Pre-Calculus Final Exam Review Spring 2014
Name:__________________________
CHAPTER 5: Trig Identities & Equations
Verify the following identities:
1.) sec  cos  sin  tan
3.)
cos 
1  sin 

1  sin 
cos 
2.) sec   sin  (tan   cot  )
4.) csc2   sec2   csc2  sec2 
5. Give the Quadrant (I, II, III, or IV) that the following angles would be found in:
a) sin x < 0 and cos x < 0
a) Quadrant ________
b) sin x > 0 and cos x < 0
b) Quadrant ________
c) sin x > 0 and cos x > 0
c) Quadrant ________
6. Solve the equations :
a). Solve from 0< x < 2 π
2 cosx + 1 = 0
7. If a =
2
3
and b =
b). Solve from 0 < x < 360
sin2x + sin = 0

find the exact values and write your answer in the blank below:
4
a). sin (a - b)
b). cos (a - b)
8. Use the angle sum or difference formulas to find the exact values
a). sin (15°)
b). cos (75°)
9. Find the exact values of sin(2x), cos (2x), and tan (2x)
When sin x =
3
4
and π < x <
(Q III)
5
2
10. Find all the exact solutions of:
sin 2x + cos x = 0
from 0 < x < 2π
Chapter 7: Systems of Equations & Matrices
11. Solve the following by substitution or elimination:
a. {
𝑥+𝑦 =8
𝑥−𝑦 =4
c. {
b. {
4𝑥 + 5𝑦 = −3
−2𝑦 = −8
d. {
5𝑥 − 𝑦 = 21
2𝑥 + 3𝑦 = −12
2𝑥 + 𝑦 = 1
4𝑥 + 2𝑦 = 3
12. Write the augmented matrix for the systems:
𝑥 − 𝑦 + 𝑧 = 10
𝑥 − 5𝑦 = 5
a. { 3𝑥 + 3𝑦 = 5
b. {
4𝑥 + 3𝑦 = 6
𝑥 + 𝑦 + 2𝑧 = 2
13. Perform the row operation on the matrix given:
1
2
-3 ⋮ -2
-5 ⋮ 5
-2R1 + R2  R2
14. Use back substitution to solve the system:
x  2 y  4z  3

 y  2z  7
z  5

15. Solve the following using row operations (put in row echelon form):
𝑥−𝑦 =6
2𝑥 + 𝑦 = −4
a. {2𝑥 − 3𝑧 = 16
b. { −2𝑦 + 4𝑧 = 0
2𝑦 + 𝑧 = 4
3𝑥 − 2𝑧 = −11
16. Solve the systems using inverse matrix (on the calculator):
2
𝑥+𝑦 =1
3𝑥 + 𝑦 − 𝑧 = 3
a. {2𝑥 − 𝑦 + 𝑧 = 1
b. {2𝑥 − 𝑦 + 𝑧 = 1
8
8
𝑥+𝑦+𝑧=3
4𝑥 + 2𝑦 = 3
17. Do the following matrix operations (if possible):
1 3 2 −1
1 2
1 6
a. [
]+[
]
c. [
]−[
]
1 3
4 7 0 8
−2 4
b. 4[
3 1
]
0 −4
d. [
2 1 1 5
][
]
0 −2 3 0
Chapter 9: Circles, Ellipses, Hyperbolas, Parametric, & Polar Equations
18. Find the standard form of the equation of the circle with the given characteristics
a. Center (4¸ -2); radius: 4
b. Center: (2, 1); point on circle: (7, 5)
19. Identify the center and radius of the circle:
a. 𝑥 2 + 𝑦 2 = 15
b. (𝑥 − 2)2 + (𝑦 + 5)2 = 12
c. (𝑥 + 5)2 + (𝑦 − 2)2 = 81
d.
𝑥 2 + (𝑦 + 5)2 = 40
20. Write the equation for the circle in standard form and identify its center and radius, then graph:
a. 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 + 9 = 0
b. 𝑥 2 + 𝑦 2 − 8𝑥 − 12𝑦 + 16 = 0
Graph the equation, identify the center, vertices, co-vertices, and foci:
23.
22.
Write an equation of the ellipse with the given characteristics and the center at (0,0):
Graph the following hyperbolas: Identify, vertices, foci, and asymptotes:
27.
Aa =
28.
.
29.
30. Write an equation for the hyperbola with a foci (0, 5) (0, -5) and vertices (0, 3) (0, -3)
31. Complete the following table for the set of parametric equations.
Plot the points (x, y) and sketch the parametric graph:
x = 3t -2
t
x
y
0
y=1–t
1
2
3
4
5
32. Use the following rectangular equation to write TWO different parametric equations: y
= 4x + 1
(Let x = t and then let x = t+1)
Parametric equations: __x=_____________
_y=___________
Parametric equations: _x=______________
_y=______________
33. Plot the given polar coordinate and find 3 additional polar representations of the points:
a. (5,

4
)
3 Points:
________________
________________
________________
a.
b. (4, 120°)
3 Points:
_________________
_________________
________________
34. Find a set of polar coordinates for the rectangular coordinate
a. (3, 4)
a. (____, ____ )
b. (4, 5)
b. ( ____, ____ )
35. Find a set of rectangular coordinates for the polar coordinate
a. (2,
b. (5,

6

4
)
a. (____ , ____ )
)
b. (____ , ____)
36. Use you calculator to graph the following
(use radians, θ min = 0, θ max = 2π, count by π/12 )
a. Sketch your graph and then
b. Label as: Limacon, Rose Curve , Circle, Archimedes Spiral, or Logarithmic
Spiral
a. r = 5
c. r = 3 – 5 sin Ө
b. r = 4sin(4θ)
d. r = 3 Ө + 2
Chapter 11: Limits & Derivatives:
37. Use the graph to determine the limit:
lim f(x)
x→ 3
38. Find : lim (x 2 + 2 x – 3)
____________________
x→2
39. Find: lim (
x 2  2 x  15
)
x5
____________________
x→ - 5
40. Find : lim 4x
____________________
x→ 3
41. Find: lim
x5
x  8 x  15
____________________
2
x→ 5
42.
Use the formula lim
f ( x  h)  f ( x )
to find the slope of the tan line for
h
f(x) = 3x 2 – 4 x
_______________
43. Find the derivative of f(x) =
x 2  4x  2
x
________________
44. Find the derivative of f(x) =
x 2  2x  1
( x  3)
________________
45. Find the derivative of f(x) =
4
_________________
x
46. Find the derivative of f(x) = (3x + 2) ( x – 1)
47. Find the derivative of f(x) =
1
x2
__________________
__________________
48. Find the derivative of f(x) = 2x 3 – 3 x 2 + 6 x – 1
__________________
49. Find the limit as x  ∞ for
f(x) =
4 x 2  3x  2
2 x 2  5x  1
__________________
50. Find the limit as x  ∞ for
f(x) =
4 x 3  5x 2  6 x
7x 2  2x
_________________
51. Find the limit as x  ∞ for
f(x) =
5x  6
2 x  3x  1
_________________
2