Download a) Write its domain in interval notation. b) Find any discontinuous

Final Review Questions
Pre-Calculus Honors
Name: __________
Date: __________
Chapter 1: Functions and Graphs
1. Find the inverse and limit domain of f(x) = -2(x + 1)2 – 3
 x  3 if x  3
2. Graph g ( x)  
 3  x if x  3
Find g(-1) and g(7).
3. Find the difference quotient for f(x) = x2 + x + 1
4. A cylindrical can is to hold one liter of oil. Express the surface area of the can, A, in cm2,
as a function of its radius, r, in cm.
1
5. Graph x 2  y 2  3x  5 y  7  0 . Give the domain and range of the graph.
2
6. For the problems below, find an equation for f 1 ( x) . Then graph f and f 1 ( x) in the
same rectangular coordinate system. State any domain restrictions necessary for f 1 ( x)
to be a function.
a. f(x) = 1 - x2
b. f(x) = x  1
7. The annual yield per walnut tree is fairly constant at 50 pounds per tree when the number
of trees is 30 or fewer. For each additional tree over 30, due to overcrowding, the annual
yield per tree for all trees on the acre decreases by 1.5 pounds.
a. Express the yield per tree, Y, in lbs, as a function of the number of trees per acre,
x.
b. Express the total yield for an acre, T, in lbs, as a function of the number of walnut
trees per acre.
c. Find the number of trees that maximize the total yield.
8. Graph f ( x) 2 x  1  1
x3  1
x2
10. An open box is made from a square piece of cardboard 24 inches on a side by cutting
identical squares from the corners and turning up the sides.
a. Express the volume of the box as a function of the length of the side of the square.
b. Find V(1), V(6) and interpret your results.
c. What is the domain of V?
d. For what length of the sides of the cut squares is the volume maximized?
9. Graph, stating all asymptotes f ( x) 
Chapter 2: Polynomial and Rational Functions
1. Find the zeros and state the multiplicity of f(x) = 3x4 + 13x3 + 15x2 – 4
2. Graph f(x) = x2(x-1)3(x+2)
x3  1
3. Graph f ( x)  3
x
4. Solve 3x4 – x3 + 4x2 – 2x – 4 = 0
5. Describe the end behavior (using arrow notation) of the function f(x) = 5x3 – 7x2 – x + 9
6. Find the remainder when f ( x)  2 x3  11x 2  7 x  5 is divided by x – 4.
7. Write the equation of a quartic polynomial with i and 2  3 as roots and passing
through (1, -6).
8. State all asymptotes and discontinuities for the following function:
x2  x  6
f ( x)  2
2 x  5x  2
9. Use the rational root theorem to find all zeros of f ( x)  x 4  6 x 3  14 x 2  14 x  5
10. You have 1000 feet of fencing to construct six adjacent corrals. Find the dimensions that
maximize the area.
11. y varies inversely with x. y = 4 when x =5. Find y when x = 2.
12. Two boys are in their bedroom by a window that is 6 m off the ground. A rocket is set
off from the ground with an initial velocity of 20 m/s. For how long will the rocket be
above their window?
13. The distance that a body falls from rest is directly proportional to the square of the time
of the fall. If skydivers fall 144 feet in 3 seconds, how long will they fall in 10 seconds?
Chapter 3: Exponential and Logarithmic Functions


1. Solve x2  3
4
3

2
3


 x2  3

5
3
0
2
3
2. Solve 2 x  13x  20  0
log 5 x
3. Solve log 4 46 
log 5 e
4. An object is heated to 100 C. It is left to cool in a room that has a temperature of 30 C.
After 5 minutes, the temperature of the object is 80 C.
a. Find a model for the temperature of the object, T, after t minutes.
b. What is the temperature of the object after 20 minutes?
c. When will the temperature of the object be 35 C?
5. Write the equation of the exponential function for which f(-1) = 3/2 and f(2) = 12.
6. Graph g(x) = log2(2 – x) + 1
3
7. Simplify

45 83n  2

4

8. Simplify 2x 2  x 2

2
9. Express y as a function of x:
1
ln xy  ln x3  4
3
10. Give the domain (in interval notation) of f ( x)  ln  x3  2 x 2  3x  6 
11. Condense into one log:
12. Graph y = -ln(2x)
x
13. Expand ln 3
e
14. Solve ln(x2 + 1) = -2
logx – log15 + log(x2 – 4) – log(x + 2)
15. Graph f ( x)  log 5 ( x  1)
16. Graph y = 3x + 1 - 2
Chapter 4: Trigonometric Functions
1. Solve the triangle: ΔABC if a = 25, b = 20 and m  B = 33
2

to x2 
3
2
A bridge has been built across a canyon. The length of the bridge is 5042 feet. From
the deepest point in the canyon, the angles of elevation to the ends of the bridge are
72º15’ and 78º20’. How deep is the canyon?
From city A to city B, a plane flies 850 miles at a bearing of N58 E. From city B to city
C, the plane flies 960 miles at a bearing of S32 E. Find the distance from city A to city
C. Then find the bearing from city C to city A.
Find the circumference of the circle if a rectangle is inscribed in a regular hexagon which
is inscribed in the circle. The length of the longer side of the rectangle is 6 in.
Using f(x) = cos x, g(x) = csc x and h(x) = tan x, find:
2. g(x) = csc x. Find the average rate of change of g from x1  
3.
4.
5.
6.
 2 3
g 1  

3 

 
b. f 1 h   
 4
7. List the domain and range of each inverse trig function.
8. An object moves in simple harmonic motion described by d  6 cos t , where t is
measured in seconds and d in inches. Find
a. the maximum displacement
b. the frequency
c. the time required for one oscillation.

 2  
9. Find the exact value of sin  cos 1 
 
 3 

10. Find the length of the arc on a circle of radius 20 feet intercepted by a 75º central angle.
Express the arc length in terms of π.
2
11. If tan    ,180     360  , find cos θ.
3
12. Find the exact values of cosA and tanA based on the information provided.
34
p < A < 2p , sec A =
16
a.


13. The wheel of a machine rotates at the rate of 300 rpm. If the diameter of the wheel is 80
cm, what are the angular and linear speeds?
14. A submarine dives at an angle of depression of 12  . If it takes 6 minutes to dive from
the surface to a depth of 500 feet, how fast (in miles per hour) does it travel along its
sloping path downward?
Chapter 4+: Trigonometric Graphs
æp
pö
j(x) = 8 - 4cos ç x - ÷
1. Graph
3ø
è6
2
3

2. Graph h( x)  1  tan  x  
3
2
2
1 5 æ 2p
pö
3. Graph h(x) = - csc ç
x- ÷
2 4 è 3
3ø
1
4. Graph y  3 tan x
2
5 æ
pö
5. Graph y = sin ç 2x + ÷
2 è
2ø
6. Graph y = -2 – sec x
7. Graph y = 3tan 1/3 x
Chapter 5: Analytic Trigonometry
sin x
1  cos x
2 cot x 

1  cos x
sin x
1. Prove:


2. Prove: sin   x   cos x
2

3. Prove:
sin      sin      2cos  sin 
4. Answer in terms of π:
3tan2x – 2secx = -4
1
5. Solve sinxcosx= 
6
2
6. Evaluate 2 cos 105  1
 7 
8 tan 

8 

7. Evaluate
 7 
1  tan 2 

 8 
 33 
8. Evaluate sin 

 8 


x 
9. Evaluate sec  sin 1 
 
2

 x  4 

sin    
 tan   tan 
10. Verify:
cos  cos 
11. Verify 4cos3  - 3cos  = cos3 
12. Find the exact value of the following:
 11 
a. cos  

 12 
b. csc(-15 )
 x  sec x  1
13. Verify tan   
tan x
2
2
14. Solve 2sin x + sinx - 2= 0
15. Solve 5cos2x - 3 = 0
16. Solve tan 2 2  4 tan 2
Chapter 6: Additional Trigonometric Topics
1. Combine forces F1 = 10 lbs, N20E and F2 = 30 lbs, N65E. Find the magnitude and
direction of the resultant force.
2. Graph r = 1 – 2sin 
3. Graph r = 4sin2 
4. Graph r2 = 9sin2 


5. Find the rectangular coordinates of  8, 
3

6. Find the polar coordinates of (-3, 5)
7. Find the dot product of v = i + 3j and w = -3i – j
8. If P1= (-1, 3) and P2 = (2, 7), write the position vector from P1 to P2 in terms of i and j
and find its magnitude, ||v||.
Chapter 9: Conic Sections and Analytic Geometry
6
1. Graph
3  6sin 
2. Eliminate the parameter and graph the plane curve.
a. x  t 2
y  t  1   t  
b. x = 1 + 3sint, y = 2cost 0,2 
3. Identify the conic section:
x2 - y2 - 2x - 2y - 1 = 0
4. Graph, giving vertices, foci and asymptotes: x 2  4 y 2  8x  2 y  7  0
5. The receiver of a parabolic television dish is 3.5 feet from the vertex and is located at the
focus. Find the equation of the cross-section of the reflector. Assume the vertex is at the
origin.
Chapter 10: Sequences and Probability
1. Find a recursive definition for 2, 6, 14, 26, 42, …
2. Write a formula for the nth term in the arithmetic sequence 14, 11, 8, … and use it to find
the value of its 23rd term.
3. Evaluate a) 12 P3 and b) 6 C 2
4. Find a formula for an: 14, 21, 2 7, 35
5. A professional basketball player signs a contract with a beginning salary of $3,000,000
for the first year and an annual increase of 4% per year beginning the second year. That
is, beginning in year 2, the athlete’s salary will be 1.04 times what it was the previous
year. What is the athlete’s salary for year 7 of the contract?
6. Evaluate without writing out the terms
i 1
1
2 

i 1  4 
7. Use the properties of sums to evaluate
5
6
i
3
i 1
8. How many numbers between 211 and 987 are divisible by 6?
9. A job pays $32,000 in the first year with an annual increase of 6% per year beginning in
the second year. What is the salary in the sixth year? What is the total salary after six
years?
10. In a geometric sequence if a2 = 5, a5 = 135, find a8.
11. Each week, the length of time it takes for college applications turned in to guidance
counselors to be reviewed by admissions officers at a college grows geometrically. An
application turned in during the second week of the school year takes 5 days to be
reviewed. One turned in during the 10th week takes 14 days. If a student waits until the
week after winter break (the 17th week of the school year), approximately how many
weeks will they have to wait for the materials to be reviewed?
10
12. Evaluate  ( k  2) 2
k 1
13. Find the probability of not picking a diamond or a face card from a standard 52-card
deck.
Chapter 11: Limits
ì 3x - 2
1. Given f (x) = í
î 3
2. Graph f(x) =
𝑥−2
𝑥 2 −6𝑥+8
x ¹ 2 find a)
lim f (x)
x®2x=2
b) lim+ f (x)
x®2
c) lim f (x)
x®2
a) Write its domain in interval notation. b) Find any discontinuous
points. Find the following limits: c) lim- f (x)
x®4
d) lim+ f (x)
x®4
e) lim f (x)
x®4
3. Determine at what numbers, if any, the following piecewise function is discontinuous.
ì
2
x£0
ï x
f (x) = í x +1 0 < x < 2
ï 5- x 2 £ x £ 5
î
4. lim
x®3
x2 - x - 6
x2 - 9
5. Find the equation of the tangent line to the graph of g(x) =
𝑥2
3
at a) x = 1 and b) x = 3.