Pre-Calculus
EOC May 16
thFinal Review Name______________________
1. A full swimming pool has 7,600 gallons in it when it begins leaking at a continuous rate of 7% per hour. a. How many gallons of water will be remaining in the pool after 5 hours? Round your answer to the nearest whole number of gallons.b. At what time t will the pool be half-full? Round your answer to three places after the decimal point.
2. Explain what you know about the end behavior, the number of Zeros (real and/or complex) and the number of relative extrema given the polynomial function:
f
(
x
)
3
x
5
2
x
4
x
2
5
3. Given the polynomial function:
f
(
x
)
2
x
3
5
x
2
x
6
, determine all the possible rational zeros (p/q): 4. Selected values for a function 𝑓(𝑥) are provided, Find 𝑓−1(3)x 𝑓(𝑥) 0 -10 1 -2 2 0 3 2 4 3
5. Determine each inverse function: 𝑓−1(𝑥). A. 𝑓(𝑥) = 40𝑥
10−𝑥 B. 𝑓(𝑥) = √𝑥 + 5
6. Determine the domain. Identify any vertical and/or horizontal asymptotes, describe the end behavior with limit notation and then sketch each graph.
a.
3
2
)
(
x
x
x
g
b.8
2
3
)
(
2
x
x
x
x
h
7. Represent the Series using sigma notation:
A.
8 + 16 + 32 + 64 + ...
B. 11 + 15 + 19 + 23 + 27 + 31 + 35
8. Expand (𝑥 + 2)49. Use polynomial long division to write
𝑥
3− 27 = (𝑥
2+ 𝑥 − 6) 𝑞(𝑥) + 𝑟(𝑥),
where q(x) and r(x) are two polynomials10. Divide:
𝑥
4+ 5𝑥
3+ 6𝑥
2− 𝑥 − 2 𝑏𝑦 𝑥 + 2
11. Simplify each rational expression:A.
x
x
x
x
x
x
2 25
24
2
2
3
24
8
9
B.3
2
2
3
2
3
4
7
2 2
x
x
x
x
x
x
C.12
7
2
3
3
2
y
y
y
y
12. Graph each exponential/logarithmic function; state the Domain & Range and Identify any Horizontal/Vertical Asymptotes:
a.
f
(
x
)
ln(
x
2
)
b.f
(
x
)
2
x
2
14. For each of the following functions: State Amplitude, Period or Wavelength, Frequency and any Horizontal and/or Vertical Translation. Identify any reflection, Vertical and/or Horizontal Stretch/Shrink. Sketch each graph over 2 fundamental periods.
A. y = 2 tan x B. y= sec (
x
)+ 1 C. y= 4 sin(x)+ 3 15. If sin 𝜃 =14 and 𝜃 lies in the second quadrant, what is cos 𝜃 equal to?16. What is the exact value of cos 5𝜋18· cos 𝜋9 + sin5𝜋18· sin𝜋9 ? 17. Evaluate each of the following:
a. arc tan (1) b. arc cos (-1/2) c. arc sin (-1/2) d. tan(arc cos (3/5))
18. Simplify: a. cot A sin A b. 2
2
sin
cos
c.
x
x
tan
2
csc
19. Verify: a.
cos
2
sec
cot
b.cos
x
sin
x
tan
x
sec
x
c. cos
u
u
2
sin
= 020. Solve for x over the interval
0,2
p
[
)
,
4
cos
2x
3
0
21. Solve for x,
0
£
x
<
2
p
,2
cos
x
sin
x
sin
x
0
22. Use the Sum/Difference formulas to determine the exact value: a.
cos 15
( )
°
b.sin 105
(
°
)
c.tan 75
( )
°
23. Given cosu=-2
,
13
12
<u<
and sin v =2
0
,
5
4
v
(Hint: Make sure to sketch a Rt. Triangle for “u” and “v”— Location is important to help determine your signs)A. Find the exact value of sin (u/2) B. Find the exact value of tan 2v
24. Identify each of the following Conic Sections, re-write each equation in standard form then sketch each graph and identify all critical points (center, vertices, minor or conjugate axis endpoints, Foci) and lines (directrix, asymptotes)
a.
6
x
2
6
y
2
12
x
36
y
36
0
b.6
x
2
9
y
2
24
x
54
y
51
0
c
4
x
2
12
y
2
4
x
96
y
155
0
d.x
2
4
x
2
y
36
0
25. Determine the equation of the parabola in standard form, given the vertex is at (-2, 5) and a solution point (-3,2). 26. Write the equation for the parabola with the given characteristics. Focus
3
,
0
, Directrix:x
3
28. A plane leaves the airport at a bearing angle of 45º traveling at 400 mph. The wind is blowing at an angle of 135º at a speed of 40 mph. What is the actual velocity of the plane?(solution video: https://www.youtube.com/watch?v=Q5e1BjuPHts )
29. Determine a unit vector in the same direction as vector U = -6i - 8 j
30. Given U = < 4, -3 > and W = < 3, 4 > determine the value of
u
w
, then Determine the angle between the vectors u and W. Include a graph.31. A point
r
,
is given in polar coordinates. A. Plot the Point and B. Convert to rectangular Coordinates
4
5
,
2
A
B
5
,
120
C
4
,
2
D
3
,
60
32. A point is given in rectangular coordinates (x, y) . Plot the point, then determine 2 different polar coordinates that would represent the same point.
A ( 2, 2) B( -1, 0 ) C.
5
2
,
5
2
33. Convert each rectangular equation to polar form: 𝐴. 𝑥 + 𝑦 = 9 B. 𝑥2+ 𝑦2= 4 34. Convert parametric equations to a rectangular form.
4
6
3
t
y
t
x
35. Rewrite each complex number in polar form: A. −6√3 + 6𝑖 B. 3 + 𝑖√3
36. Identify the function type, determine the Domain and Range (Use interval notation), determine if they are even, odd or neither and determine the intervals over which the function is increasing, decreasing, or constant.
a.
f
x
3
x
b.f
x
2
x
3
3
x
2
12
x
c.
0
,
3
0
,
3
2
x
x
x
x
x
f
37. What is the coefficient of the
𝑥
9𝑦
term in the expansion of(𝑥
3+ 5𝑦)
4?
38. Given f(x) and g(x), Evaluate (f-g) (x) and (f/g) (x) then determine the domain of each new combination.
x
x
g
x
x
f
2
2
3
,
5
39. Determine the Domain for f(x) and g(x). Determine
f
g
x
. Show your work, then determine your new Domain
x
x
2,
g
x
x
3
f
40. A punter kicks a football from an initial height of 2 feet with an initial velcocity of 82 feet per second at an angle of 63º with the ground. Use parametric equations to model the motion of both the horizontal and vertical positions, with respect to “t”, time in seconds. What is the position of the ball at 1.5 seconds?
