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Pre-Calculus Final Review Packet
Name_________________________________
Rules for the UW Math 120 Final
•
You get one 8.5” x 11” sheet of handwritten notes
•
You can only use a scientific calculator
•
7-9 questions in 170 minutes (it works out to 19-24 minutes per problem)
•
Has to be taken in one sitting (no extra time)
Table of Contents of this Packet
Page
2
Summary Topics for the Final Exam
Ø This is UW’s actual review packet given to their students. They only list topics.
4
Parametric and Circular Equations
Ø Common themes: Parametric equations of motion; finding time 2 equations are closest
together; Circle equations with D = RT
6
Writing Quadratic, Linear-to-Linear, and Exponential Equations
Ø Common themes: Algebraically find each type of equation from a given set of points
8
Multipart Functions, Inverses, and Composite Functions
Ø Common themes: Complicated algebraic manipulation
10
Circular Motion, Belt and Wheel Problems
12
Sinusoidal Functions and Triangle Trig
Ø Common themes: Write equation of sinusoidal from given max & min; use the inverse
14
Answers to all problems in this packet
15
Calendar for the rest of the year
Your final exam will be:
Friday, June 2 from 7:50am – 10:40am
NOTE: In addition to problems from this packet, you should also go to
http://www.math.washington.edu/~m120/testindex.php
to find more problems to work on.
Good Luck! J
UW Pre-Calc Final Review Page
1
Summary Topics for the Final Exam – UW Math 120
[Page 2-3 was the actual entire final review packet handed out to a UW Pre-Calculus class during Winter 2012.]
Here is a summary of the topics for the final exam. The core of your studying should be the assigned
homework problems: make sure you really understand those well before moving on to other things (like the
old midterms on the test archive). In the test archive, there are many old final exams for you to study with.
Every topic on old finals is fair game for your upcoming final exam.
Chapter 1 - Warm Up
– One of the most important ideas of this chapter is that of multiplying by one as a means of unit
conversion. This idea makes all unit conversions have a common method, and helps one’s note keeping.
Chapter 2 - Imposing Coordinates – This chapter introduced the use of the coordinate system and the
distance formula.
– A classic problem from this chapter is one in which two objects are moving and we need to describe the
distance between them, like problems 2.3, and 2.10.
Chapter 3 - Three Simple Curves
– This chapter introduces circles and horizontal and vertical lines. You should be sure you are comfortable
finding the equation of a circle from a variety of descriptions.
– You should be able to find the intersection of a circle with a vertical or horizontal line.
Chapter 4 - Linear Modeling – In this chapter, we get the general line definition. Be sure you are able to
find the intersection of a given circle with a general line.
– We also have the idea of perpendicular lines, and the method for finding the shortest distance between a
line and a point not on that line. We also considered tangent lines to circles.
– Uniform linear motion is introduced. See problems 4.13 and 4.14.
– Especially good problems are 4.6, 4.7, 4.8, 4.10, 4.11.
Chapter 5 - Functions and Graphs – Here the function is introduced.
– Every function has a domain, range and graph. Be sure to know what each is, and how to determine it
for a given function. As we said, finding the range and graph can be hard; rest assured, if asked to find
the range or graph of a given function, it will be doable.
– You should be comfortable with multipart functions (what are they, how to evaluate one, how to solve
equations involving them, etc.) What’s an example of a multipart function?
Chapter 6 - Graphical Analysis
– Chapter 6 talks about a variety of function-related topics.
– You should understand how to graph a multipart function, where each part is linear.
– Especially good problems are 6.4, 6.6, 6.8, and 6.9.
Chapter 7 - Quadratic Functions
– You should know that quadratic functions are those of the form f(x) = ax2 + bx + c and that these can
always be put into vertex form f(x) = a(x − h)2 + k. You should be able to find the vertex of a quadratic
function.
– You should be able to create quadratic models given three points, or the vertex & one other point.
– You should be able to find the maximum or minimum value of a quantity determined by a quadratic
function by considering the vertex.
– I like problems 7.9-7.17 a lot.
Chapter 8 - Composition
– You should know what it means to compose two functions. You should understand what is meant by
f(g(x)). You should know that f(g(x)) and g(f(x)) are generally different functions. You should be able to
write simplified rules for compositions f(g(x)) and g(f(x)) given rules for f(x) and g(x).
– I particularly like problems 8.2, 8.3 and 8.4.
Chapter 9 - Inverse Functions
– You should understand what an inverse function is, what conditions a function must satisfy in order to
have an inverse (do all functions have inverses? can you tell if a function has an inverse by looking at its
UW Pre-Calc Final Review Page 2
graph?), and how to find the inverse of a given function
– You should understand what a one-to-one function is, and what is special about the graph of a one-toone function.
– I like problems 9.2, 9.5, and 9.7.
Chapters 10, 11, 12 - Exponential functions, modeling and logarithms
– You should be able to recognize functions of the form f(x) = A0bx or, equivalently, f(x) = A0ekx. You
should be able to put exponential functions into these forms.
– You should be able to create exponential models of quantities that change over time. Given two values
of the quantity at two data points in time, you should be able to come up with an exponential model that
fits the data. Given a single data point and information about the quantity’s rate of growth (e.g.,
percentage annual increase, or doubling time), you should be able to write an exponential model that fits.
– You should be able to solve equations involving exponential functions using the natural logarithm.
– Relevant problems include 11.1, 11.2, 12.7, 12.9, 12.10, 12.11, 12.12.
Chapter 13 - Three Construction Tools
– You should understand horizontal & vertical shifting, and horizontal and vertical scaling (aka dilating).
– You should understand how to derive the graph of g(x) = af(bx+c)+d from the graph of f(x) (see #13.2)
– I especially like problems 13.2, 13.3, and 13.5.
Chapter 15 - Measuring an Angle
– You should understand how to convert between degrees and radians
– You should understand and be able to use the relationships between radii, angle, arc length and area
– I like problems 15.8 and 15.9.
Chapter 16 - Measuring Circular Motion
– You should understand the various measures of angular speed (aka angular velocity), like rpm, radians
per second, or degrees per hour
– You should understand the relationship between radius, angular speed and linear speed
– You should know how solve a belt-and-pulley problem (e.g., the bicycle example from lecture, example
16.4.1, problems 16.2, 16.7 and 16.8) – I like problems 16.2, 16.3, 16.5, 16.7, 16.8.
Chapter 17 - The Circular Functions
– This chapter introduces the trigonometric functions.
– You should be able to solve problems using the idea of trigonometric functions as ratios of sides of right
triangles (e.g., problems 17.2, 17.3, 17.7, and 17.8) and some algebra
– You should understand the definitions of sin x and cos x using the unit circle; you should be able to
determine certain simple properties of the functions sin x and cos x from this definition (e.g., the range,
the domain, the graph, the values at certain value of x, like x = 5π/2)
– You should be able to determine the location of an object moving circularly given information about its
speed and starting location (e.g., problems 17.1, 17.4, 17.5, 17.6, 17.9, 17.10, 17.12, 19.6, 20.8)
Chapter 18 - Trigonometric Functions
– This is a short chapter which adds some final touches to our knowledge of the functions sin x and cos x
and related functions.
– You should be thoroughly familiar with the graphs of y = sin x and y = cos x
Chapters 19, 20 - Sinusoidal Functions
– You should understand the idea of a sinusoidal function as a shift/dilated version of the function sin x.
– You should understand the effect of the four parameters A, B, C and D on of f(x) = A sin B(x − C) + D
– You should be able to model with sinusoidal functions. In particular, you should be able to determine
the parameters A, B, C, and D from a verbal description of a quantity that varies sinusoidally with time
(e.g., problems 19.2, 19.3, 19.4)
– You should be able to solve equations of the form f(x) = k where f is a sinusoidal function; if there are
any solutions, there are infinitely many, and you should be able to find them. You should be able to do
this in the context of a modeling problem (e.g., problems 20.3, 20.4, 20.5, 20.6, 20.7)
UW Pre-Calc Final Review Page
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Parametric and Circular Equations
Things you may want to write down on your cheat sheet:
Distance Formula: d =
2
( x2 − x1 ) + ( y2 − y1 )
Equation of a Circle: (x – h) + (y – k) = r
2
2
2
2
Min or Max of a Quadratic: x =
−b
2a
Eq. of a Line: y – y = m(x – x ) where m =
1
1
y2 − y1
x2 − x1
EXAMPLE PROBLEM
W2012 #1) Rudy and Murray are moving in the xy-plane along straight lines at constant speeds.
Rudy starts from the point (10, 0) and heads directly toward the point (−6, 6), reaching it in 10
seconds. Murray starts from the point (3, 7) and travels along the line y = x + 4. Murray moves
toward the y-axis, and takes twice as long to reach the y-axis as Rudy does to reach the y-axis.
(a) Find the parametric equations of motion for Rudy.
(b) Find the parametric equations of motion for Murray.
(c) How long has Rudy been moving when the distance
between Rudy and Murray is as small as it ever gets?
EXAMPLE PROBLEM
W2012 #2) Inga is walking near the Circular Forest, a forest in the shape of a circle with a radius of
10 km. Inga starts her walk from a point 13 km WEST and 3 km NORTH of the center of the forest.
She starts walking due east. After reaching the forest, she continues traveling east for 5 km, and then
turns and walks due SOUTH until leaving the forest. Inga walks at a constant speed of 3 km per
hour. How much time did she spend in the forest during her walk?
UW Pre-Calc Final Review Page
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Parametric and Circular Equations (cont.)
HOMEWORK PROBLEMS (do on a separate piece of paper)
W2011 #4) Amy and Hans are moving at constant speeds along lines in the xy-plane. Amy starts at
(0, 4) and moves along the line y = 4 − 0.2x. She will reach the x-axis in 5 seconds. Hans starts at the
same time as Amy. He starts from the origin, heads directly toward the point (8, 12), and will reach
it in 4 seconds.
(a) Give Amy’s equations of motion (i.e., her x- and y-coordinates as functions of time, t).
(b) Give Hans’ equations of motion (i.e., his x- and y-coordinates as functions of time, t).
(c) When will Amy and Hans be closest together?
W2011 #5) Bob is 40 km east and 18 km north of a radio tower. Bob walks in a straight line toward a
point 30 km due west of the tower, and then continues in the same direction on to his destination.
How close does Bob come to the radio tower on his walk?
A2011 #5) William took a walk near the Circular Dunes. The Circular Dunes is a perfect circle, with
a radius of 7 km. William began his walk from a point 11 km due north of the center of the Dunes.
He walked due east for one hour, and then due south for three hours. He then walked due west
until he left the Dunes. William walked at a constant speed of 3 km/hr.
(a) For what length of time was William in the Dunes?
(b) Suppose William had walked in a straight line from the point where he entered the Dunes
to the point where he exited. If he continued along that line, how far west of the center of the
Dunes would he be when he was due west of the center?
S2011 #1) Toshiro is walking near the Circular Forest, which has the shape of a perfect circle, and
radius of 8 km. He begins from a point 10 km WEST and 3 km SOUTH of the center of the forest. He
heads directly toward a point 20 km EAST and 4 km NORTH of the center of the forest. However,
when he reaches a point due EAST of the center of the forest, he turns and walks due SOUTH until
he leaves the forest. Toshiro walks at a constant 5 km per hour.
How much time did he spend in the forest?
A2010 #1) A radar buoy detects any boats within a radius of 12 miles. A tugboat starts at a location
18 miles SOUTH and 10 miles EAST of the radar buoy. The tugboat travels at a constant speed of 15
mph. The tugboat travels on a straight line toward the NORTHERNMOST point of the radar region.
When the tugboat is directly EAST of the buoy, it turns and travels DUE NORTH until it exits the
radar region.
How long (in hours) is the tugboat in the radar region?
UW Pre-Calc Final Review Page
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Writing Quadratic, Linear-to-Linear (Rational) and
Exponential Eqns.
Things you may want to write down on your cheat sheet:
Standard Form Quadratic: f(x) = Ax + Bx + C
2
Vertex at: x =
−B
2A
Standard Exponential: f ( x ) = a ⋅ b x
Vertex Form Quadratic: f(x) = a(x – h) + k
2
Switching Standard to Vertex: A = a B = –2ah C = ah + k
2
Standard Lin-to-Lin: f ( x ) =
ax + b
where a = HA & –c = VA
x+c
EXAMPLE PROBLEM
W2012 #6) Margarita knows that her score on the final exam will be a quadratic function of the
amount of time she studies. If she studies for 10 hours, she will score 50 %. If she studies for 20
hours, she will score 85 %. If she studies for 35 hours, she will score 80 % (her score decreases due to
overwork and stress from lack of sleep).
(a) How much should she study to maximize her score on the exam?
(b) What is her maximum possible score on the exam?
EXAMPLE
A2011 #7) The population of the city of Alk increases by 17 percent every 12 years. In 2010, the
population of Alk was 8,000. The population of the city of Bem doubles in the length of time it
takes for the city of Alk to triple. In 2005, there were 15,000 people in Bem. When will the cities have
the same population? Give your answer in years after 2010.
EXAMPLE
S2011 #5) A tree is growing. Its height is a linear-to-linear rational function of time. Today, the tree
is 5 feet tall. Twenty years from now it will be 40 feet tall, and 21 years from now it will be 41 feet.
(a) Express the tree’s height as a linear-to-linear function of t, where t is years from today.
(b) When (in years from today) will the tree be 50 feet tall?
UW Pre-Calc Final Review Page
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Writing Quadratic, Linear-to-Linear (Rational) and
Exponential Eqns. (cont.)
HOMEWORK PROBLEMS (do on a separate piece of paper)
A2011 #1) You have 300 meters of fencing with which to build two enclosures. One will be a square,
and the other will be a rectangle where the length of the base is exactly twice the length of the
height.
(a) Give the dimensions of the square and rectangle that minimize the combined area.
(b) What is the maximum combined area?
W2011 #1) A producer is figuring out how much to charge for tickets to a show. If she charges $0
per ticket, she will make $0. If she charges $5 per ticket, she will make $1775. If she charges $15 per
ticket, she will make $975. If the amount of money she makes is a quadratic function of the ticket
price, what is the maximum possible amount of money she can make from the sale of the tickets?
A2010 #4) The value of Goofy’s dog house started to decline at the beginning of 2008 according to a
quadratic model. At the start of 2008, his house is worth $300. At the start of 2009, the value is $240.
At the start of 2013, the value will be $260.
(a) What is the lowest value that the house reaches? (Round to the nearest cent).
(b) At the beginning of 2013, the value of the house starts to grow according to an
exponential model. After that time, the value of the home doubles every 30 years. In what
year will the value of the home reach $1,000? (Give your answer as the year with two digits
after the decimal).
S2011 #6) The population of city A increase by 2.38% per year. City B doubles in the length of time
it takes city A to increase from a population of 4,000 to a population of 9,500. In the year 2000, city B
had a population of 11,000.
When will city B’s population reach 30,000? Express your answer in years after the year 2000.
A2011 #6) Rosetta is growing a bamboo plant in her apartment. The height of the plant is a linear-tolinear function of time. Thirty days ago, the plant was 14 cm high. Today, the plant is 18 cm high.
The plant always increases in height, and will approach (but never exceed) a height of 32 cm.
(a) Find a function representing the height of the plant as a function of time.
(b) Rosetta also has a fast-growing cactus. Today, its height is 9 cm. The cactus grows at a
constant rate of 1 cm per day. When will the cactus and the bamboo plant be the same
height? Give your answer in days after today.
W2011 #6) The more hours Matt puts into his training, the faster he will be able to complete an upcoming 200 km bicycle ride. If he puts in 30 hours of training, he will be able to complete it in 9
hours. If he trains for 40 hours, he will be able to complete the ride in 8 hours. No matter how much
he trains, he will never be able to complete the ride in less than 6 hours. Assume that the time Matt
will take to complete the ride is a linear-to-linear rational function of the time he spends training.
If Matt wants to finish the ride in 8.5 hours, how much training should he do?
UW Pre-Calc Final Review Page
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Multipart Functions, Inverses, and Composite Functions
Things you may want to write down on your cheat sheet:
EXAMPLE PROBLEM
W2012 #7) Margot has a pizza in the shape shown below. The dimensions are in centimeters.
Margot wants to make a vertical cut in the pizza. If her cut is x centimeters from the left edge,
express the area to the left of the cut as a multipart function of x.
EXAMPLE PROBLEM
W2012 #8) (a) Let h(x) =
(b) Let k(x) =
2
x +
5
x + x −1 . Find h −1 (x) .
x . Find the fixed points of k(x). (Rietman note: This means where are the x and
y coordinates the same?)
UW Pre-Calc Final Review Page
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Multipart Functions, Inverses, and Composite Func. (cont.)
HOMEWORK PROBLEMS (do on a separate piece of paper)
S2011 #7) Let f(x) = x + 3 x +1 , and g(x) = 2x+4.
(a) Express f(g(x)) as a multipart function.
(b) Find all solutions to the equation f(g(x)) = 12x.
A2010 #8)
(a) Let f(x) =
3x +1
2
and g(x) =
. Let h(x) = f(g(x)). Find h −1 (x) .
x−2
x −1
(b) Let k(x) = 2x 2 − 4x + 7 restricted to x ≤ 1. Find k −1 (x) .
W2010 #2) A polygonal pizza is shown below, with the horizontal and vertical side lengths labeled
in inches.
If a vertical cut is made x inches from the left edge of the pizza, what is the area to the left of the cut?
Express the area as a multipart function of x.
W2010 #3) Julia is selling tickets to a concert. She knows from previous experience that the number
of tickets she will sell depends on the price she charges per ticket. The relationship is given by
#
100
%500 −
p if 0 ≤ p ≤ 12
f ( p) = $
3
%&250 −12.5p if 12 ≤ p ≤ 20
where p is the price per ticket (in dollars) and f(p) is the number of tickets sold.
(a) Let M(p) be the amount of money Julia will make from the sale of tickets at price p. Write the
multi-part rule for the function M(p).
(b) Sketch a graph of M(p).
(c) What price p should she charge to make the most money?
UW Pre-Calc Final Review Page
9
Circular Motion, Belt & Wheel problems
Things you may want to write down on your cheat sheet:
EXAMPLE PROBLEM
W2012 #4) Maria is running around a circular track. She runs clockwise and, from when she starts, it
takes her 28 seconds to reach the southernmost point of the track. She takes 105 seconds to run each
lap of the track.
(a) From when she starts, how long does it take her to reach the easternmost point of the track?
(b) Bob starts running on the track at the same time as Maria. He starts from the northernmost point
and runs counter-clockwise. He takes 130 seconds to run each lap of the track. Let R be the radius of
the track. Using a coordinate system with the origin at the center of the track, find Bob’s x- and ycoordinates when he passes Maria for the first time (your answer will involve R).
EXAMPLE PROBLEM
A2010#3) The treadmill at Felix’s gym is run by a belt and wheel system. A motor spins wheel A
which is attached by a belt to wheel B. Wheel B is fixed by an axle to wheel C which spins the
conveyor that Felix runs on. The machine is set to run the conveyor at a pace of 1 mile every 8
minutes.
The radius of wheel A is 0.9 inches, the radius of wheel B is 0.75 inch, and the radius of wheel C is 3
inches. At what angular speed is the motor rotating Wheel A? Give your final answer in revolutions
per minute.
UW Pre-Calc Final Review Page
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Circular Motion, Belt & Wheel problems (cont.)
HOMEWORK PROBLEMS (do on a separate piece of paper)
W2010 #5) You are designing a system of wheels and belts as pictured below. You want wheel A to
rotate 11 RPM while wheel B rotates 37 RPM. Wheel A has a radius of 6 inches, wheel B has a radius
of 7 inches and wheel D has a radius of 4 inches. Assume wheels C and D are rigidly fastened to the
same axle. What is the radius r of wheel C?
A2011 #8) Maria is riding a ferris wheel. Her linear speed is 5 meters per second. After the ride
starts, it takes Maria 8 seconds to reach the highest point on the ride. It takes 18.5 seconds from
when the ride starts for Maria to reach the lowest point on the ride. The highest point of the ride is
36 meters off the ground. How high above the ground is Maria 120 seconds after the ride starts?
W2011 #3) Kurt and Lotte are running around a circular track at constant speeds. They start running
at the same time. Kurt runs clockwise, and starts from the northernmost point. He takes 11 seconds
to reach the easternmost point for the first time. Lotte runs counterclockwise, starting from the
westernmost point. She passes Kurt for the first time after 19 seconds.
(a) How long does Lotte take to complete one lap of the track?
(b) Suppose the track has a radius of 20 meters. Find the distance between Lotte and Kurt
after they have been running for 25 minutes.
S2011 #2) Arnoldo and Hamda are running around a circular track. Arnoldo starts from the
northernmost point of track and runs clockwise. Arnoldo takes 23 second to run each lap of the
track. Hamda runs counterclockwise, and takes 27 seconds to run each lap of the track. Arnoldo and
Hamda start running at the same time, and pass each other for the first time after 8 seconds.
(a) How long have Arnoldo and Hamda been running (i.e., time since they started running)
when they pass each other for the second time?
(b) Let R be the radius of the track. With the origin at the center of the track, express
Hamda’s x- and y-coordinates as functions of the time, t, since Hamda started running (your
answers will involve R).
A2010 #2) Thelma and Louise are each going for a jog around a circular track of radius 80 feet. They
start jogging from different locations on the circle. Thelma starts at the EASTERNMOST point and
jogs 10 feet/second around the track in the clockwise direction. Louise runs counterclockwise at 14
feet/second and passes Thelma for the first time in 16 seconds.
Find Louise’s x and y coordinates in 12 minutes.
UW Pre-Calc Final Review Page
11
Sinusoidal Functions and Triangle Trig
Things you may want to write down on your cheat sheet:
30-60-90 (general):
30-60-90 (unit circle):
30°
30°
3
2
2x
x 3
€
Trig Ratios:
O
O
H
A
cos ( B) =
H
O
B tan ( B ) = A
sin ( B) =
H
A
45-45-90 (unit circle):
2
2
s 2
45°
1
s
1
2
x
45°
1
60°
60°
45-45-90 (general):
H
O
H
sec ( B) =
A
A
cot ( B) =
O
csc ( B) =
45°
€
s
Inverse domains:
π
π
≤θ ≤
2
2
Cos: Quad I & II, 0 ≤ θ ≤ π
π
π
Tan: Quad I & IV, − ≤ θ ≤
2
2
Sin: Quad I & IV, −
45°
2
2
Sinusoidal:
" 2π
%
y = Asin $
( x − C )' + D
# per
&
A = Amplitude
C = Phase Shift
D = Mean Value
EXAMPLE PROBLEM
W2010 #7) You are measuring trees in a forest. Standing on the ground exactly halfway between
two trees, you measure the angle the top of each tree makes with the horizon: one angle is 67 , and
the other is 82 . If one tree is 80 feet taller than the other, how far apart (horizontal distance along the
ground) are the two trees?
o
o
EXAMPLE PROBLEM
A2011#3) A weight is attached to a spring suspended from the ceiling. The height h(t) of the weight
is a sinusoidal function of time t. At time t = 5 seconds, the weight is at its lowest height of 15 cm.
The weight next reaches its highest height of 37 cm at time t = 9.4 seconds. During the first 20
seconds, how much time is the weight above 28 cm?
UW Pre-Calc Final Review Page
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Sinusoidal Functions and Triangle Trig (cont.)
HOMEWORK PROBLEMS (do on a separate piece of paper)
S2011 #3) The volume of a certain weather balloon is a sinusoidal function of time. At 1 AM today,
the volume was at a minimum, 2 m . The volume then increased, reaching a maximum of 22 m at 5
AM today.
3
3
(a) Express the volume of the balloon as a sinusoidal function of time, t, where t is hours after
midnight today.
(b) In the first 14 hours after midnight, for how much time was the balloon’s volume less
than 6 m ?
3
W2011#2) The pressure inside an artery is varying sinusoidally. With a heart beat at a rate of 60
beats per minute, the period of this sinusoidal function is 1 second. The maximum pressure is 16,000
pascals, and the minimum pressure is 11,000 pascals. At time t = 0, you begin to measure the
pressure. A maximum is attained for the first time at t = 0.23 seconds.
Between t = 0 and t = 1.3, how much time (in seconds) is the pressure above 15,000 pascals?
W2011 #7) The crew of a helicopter needs to land temporarily in a forest. They spot a flat horizontal
piece of ground (a clearing in the forest) as a potential landing site, but are uncertain whether it is
wide enough. They make two measurements from point A (see figure) and find α=19 and β=60 .
They then rise vertically 100 feet to point B and measure γ = 53 . Determine the width of the clearing.
o
o
o
A2011 #2) Erika is measuring the height of a tree. She is standing on the ground at some distance
from the tree and measures an angle of 63 degrees to the top of the tree. She walks 20 feet further
away from the tree and measures an angle of 50 degrees to the top of the tree (this is her second
measurement). A year later she comes back to check on the tree and it has grown. She measures the
angle from the same location as the second measurement from the year before and now gets an
angle of 52 degrees.
How much taller is the tree?
W2012 #5) The pressure inside Simone’s eyeball is a sinusoidal function of time. The pressure
reaches a minimum of 1.8 kPa 4 hours after you begin measuring it. The pressure then increases,
reaching a maximum of 2.3 kPa 9 hours after you begin measuring it.
How long do you have to measure the pressure until you have observed it below 1.9 kPa for
5 hours?
UW Pre-Calc Final Review Page
13
ANSWERS – Parametric & Circ.
EXAMPLE ANSWERS
W2012 #1) (a) (10-1.6t, 0.6t)
(b) (3-0.24t, 7-0.24t) (c) 6.025 seconds
W2012 #2) 5.64 hours
HOMEWORK ANSWERS
W2011 #4) (a) (4t , 4-0.8t)
(b) (2t , 3t)
(c) 0.8243 seconds
W2011 #5) 7.47km
A2011 #5) (a) 4.68 hours
(b) 11.2 km
S2011 #1) 3.65 hours
A2010 #1) 1.43 hours
ANSWERS –Quad., Rat., & Exp.
EXAMPLE ANSWERS
W2012 #6) (a) 26.41304 hours
(b) 91.306%
A2011 #7) 138.727 years after 2010
S2011 #5) (a) h(t) = (89t+140)/(t+28)
(b) 32.30769 years from today
HOMEWORK ANSWERS
A2011 #1) (a) Square side length 600/17, and
rectangle height 450/17 and width 900/17
(b) 5625 square meters
W2011 #1) $2155.17
A2010 #4) (a) $197.52
(b) 2071.30
S2011 #6) 53.23064 years after the year 2000
A2011 #6) (a) h(t) = (32t+2430)/(t+135) (today
is t = 0)
(b) 9.96211 days after today
W2011 #6) 34 hours
ANSWERS – Multipart, Inv, Comp
EXAMPLE ANSWERS
W2012 #7)
)1 "
%
8
if 0 ≤ x ≤ 7
+ x $17 + ( x − 7) + 25'
+2 #
&
7
Area(x) = *
+ 7 17 + 25 + 1 x − 7 " 25 − 20 x − 36 + 5% if 7 < x ≤ 36
) ( ) $#
(
) '&
+, 2 (
2
29
)1 !
$
8
if 0 ≤ x ≤ 7
+ x #17 + ( x +17) &
%
7
+2 "
Area(x) = *
+147 + 1 x − 7 ! 25 + ! − 20 x + 865 $$ if 7 < x ≤ 36
&&
( ) # #"
+,
2
29
29 %%
"
W2012 #8) (a) h−1 (x) = 1 x 2 + 1 + 1
4
2 4x 2
OR
(b) 0 and 25/9
HOMEWORK ANSWERS
S2011 #7) (a)
#
5
%%8x +19 if x ≥ −
2
f ( g ( x )) = $
%−4x −11 if x < − 5
%&
2
(b) 19/4 is the only solution
A2010 #8) (a) h−1 ( x ) = 5 − 4x
−2x −1
W2010 #2)
OR
(b) k −1 ( x ) = 1− x − 5
2
#
% 4x
if 0 ≤ x ≤ 3
%
A ( x ) = $12 + 6 ( x − 3)
if 3 ≤ x ≤ 4
%
1
%18 + ( x − 4) ( 6 − 3 ( x − 6 )) if 4 ≤ x ≤ 6
&
2
#
% 4x
if 0 ≤ x ≤ 3
%
A ( x ) = $12 + 6 ( x − 3)
if 3 ≤ x ≤ 4
%
1
%18 + ( x − 4) ( 6 + (−3x +18)) if 4 ≤ x ≤ 6
&
2
) "
100 %
W2010 #3) (a) M p = +* p $# 500 − 3 p '& if 0 ≤ p ≤ 12
( )
+ p 250 −12.5p
)
, (
if 12 ≤ p ≤ 20
(b)
(c) She should charge $7.50 per ticket to make
the most money.
ANSWERS – Circ Mtn, Belt & Whl
EXAMPLE ANSWERS
W2012 #4) (a) 1.75 seconds
(b) (−0.8356238R, −0.549302R)
A2010 #3) 350.14 revolutions per minute
HOMEWORK ANSWERS
W2010 #5) 1.0193 inches
A2011 #8) 10.933096 meters
W2011 #3) (a) 59.714 sec/rev (b) 39.693 m
S2011 #2) (a) 20.42 seconds
(b) In radians: x = R cos(−2.4763439676 + 2"t)
y = R sin(−2.4763439676 + 2"t)
In degrees:
x = R cos(218.116 + 13.33t)
y = R sin(218.116 + 13.33t)
A2010 #2) (−19.690202, 77.538996)
ANSWERS – Sinusoidal & Tri Trig
EXAMPLE ANSWERS
W2010 #7) 33.62 feet apart
A2011 #3) 10.1757 seconds
HOMEWORK ANSWERS
S2011 #3) (a) V (t) = 10sin "$ 2π (t − 3) %' +12
# 8
&
(b) 4.5420068 hours
W2011 #2) 0.5128 seconds
W2011 #7) 282.6937 feet
A2011 #2) 4.490555 feet
W2012 #5) 14.57249 hours
UW Pre-Calc Final Review Page
14
Per
6
Th 5/14
(B)
M 5/18
(C)
Tu 5/19
(B)
Th 5/21
(B)
M 5/25
Tu 5/26
(B)
DUE at the
BEGINNING of class
today
UWPC Homework
Assignments are listed by the date they are due.
Class activities are listed on the day they take place.
REVIEW FOCUS TODAY: Parametric Equations & Circle Equations
All problems on p5
Practice for the Graphing Mastery
Mastery Practice
Worksheet
SBA Test: 65 minute classes
REVIEW FOCUS TODAY: Quadratic, Linear-to-Linear & Exponential
Functions
All problems on p7
REVIEW FOCUS TODAY: Multipart, Inverse, & Composite Functions
1st MASTERY ATTEMPT
NO SCHOOL – MEMORIAL DAY
All problems on p9
REVIEW FOCUS TODAY: Circular Motion (Belt & Wheel problems)
2ND MASTERY ATTEMPT
MOCK FINAL WED 2:50-5:40 pm Library
W 5/27
Th 5/28
(B)
All problems on p11
MOCK FINAL THU 2:50-5:40 pm Library
MOCK FINAL SAT 10:00am-12:50pm Library
Sat 5/30
M 6/1
(C)
Tu 6/2
(B)
This will count for 3 assignments and a big chunk of participation points. You
need only attend one…it's the same practice exam!
REVIEW FOCUS: Sinusoidal Func. & Triangle Trig
3rd MASTERY ATTEMPT
All problems on p13
If you’ve already attended the Wednesday or Thursday session, a second
practice final will be available on Saturday (optional, NOT for extra credit)
Annual day – 40 minute classes
Grade your mock final according to the UW grading system
FINAL EXAM TODAY @ 7:50 a.m.
Don’t forget your cheat sheet & scientific calc.!
In Class (50 minutes): Debrief the final
Th 6/4
(B)
M 6/8
Tu 6/9
(B)
Th 6/11
(B)
Tu 6/16
(B)
Th 6/18
(B)
Fri 6/19
(C)
NO SCHOOL - TEACHER WORK DAY
Senior Finals
SENIORS: GRAPHING MASTERY DUE
80 minute classes, Dismiss at 12:20
NON-SENIORS: GRAPHING MASTERY DUE
24 minute classes, Dismiss at 10:40
UW Pre-Calc Final Review Page
15