precalculus exam questions modules 2-3
TRANSCRIPT
Precalculus Exam Questions: Modules 2-‐3
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Test Instructions 1. Please read each problem carefully.
2. Fill out Scantron with your information. Be sure to include your ID Number.
3. Mark all answers clearly on the scantron sheet provided.
4. For open-ended questions, answers without supporting work will be given zero
credit. Partial credit is granted only if work is shown. 5. No calculators with Qwerty keyboards or ones like the Casio FX-2, TI-89 or TI-92
that do symbolic algebra may be used. 6. Proctors reserve the right to check calculators.
7. The usage of cell phones is prohibited. TURN YOUR CELL PHONE OFF! Do not
allow your cell phone to ring while you are taking the exam. Do not use the calculator on your cell phone. If a proctor sees you using a cell phone, they will take your exam away from you.
Multiple Choice Section (Worth 3 points each) – Mark all answers on this exam. Show work on this exam to receive partial credit for incorrect answers.
1. A 15-‐inch candle burns 1.2 inches per hour. If h = the number of hours that the
candle has been burning and L = 1.2 h, what does the variable L represent in this formula? a. L = the rate at which the candle is burning after h hours b. L = the final length of the burning candle in inches c. L = the length of the candle in inches that remains after h hours d. L = the number of inches that has burned from the candle after h hours e. L = the length of the part of the candle that has burned (in inches) when it
stops burning 2. Millie ran at a constant speed of 3 miles per hour as she ran 4 laps on a track.
Which of the following describes a varying quantity in this situation? a. The distance of one lap b. The elapsed time since Millie began running c. The number of laps Millie had run when she was finished running d. The speed that Millie was running e. The time it took Millie to run one lap.
3. Suppose r changes at a constant rate of 2.5 with respect to p. What does this mean for
any change in p?
a. The change in p is increasing at a constant rate of 2.5 b. The change in r is always 2.5 times more than the change in p c. The change in p is always 2.5 times more than the change in r
d. The change in r is always 2.5 times as much as the change in p e. The change in p is always 2.5 times as much as the change in r
4. Jim rides his bike at a constant rate of change 21.5 miles per hour as he travels from his
home to a coffee shop. Read all five options and select the one that includes all correct answers.
a. The number of miles Jim travels is always 21.5 times as large as the number of hours Jim
rides. b. If Jim rides ½ of one hour he will ride ½ of 21.5 miles. c. For every 1 hour of elapsed time Jim travels a distance of 21.5 miles. d. All of the above e. A and C only
5. After completing a pit stop, a racecar pulls back onto the racetrack. Let f (t) be the distance of the racecar t seconds since leaving the pit. What is the meaning of f(5) – f(3)? a. The distance traveled by the car from t = 3 to t = 5 seconds since leaving the
pit b. The average speed of the car from t = 3 to t = 5 seconds since leaving the pit c. The increase in speed as the car moved from the 3 m mark to the 5 m mark d. The average speed of the car as the car moved from the 3 m mark to the 5 m
mark e. The time elapsed as the car moved from the 3 m mark to the 5 m mark
(Use the following statement to answer questions 6 & 7) Cameron cycles along a stretch of a cycling path at a constant rate of 23 ft/sec. 6. How many feet does Cameron travel in 4.3 sec?
a. 5.35 ft b. 18.7 ft c. 27.3 ft d. 98.9 ft e. None of these
7. Which formula determines Cameron’s distance d from a milepost (in feet), given the amount of time t (in seconds) since Cameron passed the milepost?
a. d= t / 23 b. t = d + 23 c. d = 23t d. t = 23d e. d = t + 23
8. A spherical snowball originally has a radius of 8 cm. As it melts the radius decreases at a rate of 0.6 cm/minute. Which of the following defines the volume of the snowball (in cm3) as a function of the amount of time (in minutes) since the snowball began melting? (Recall that the volume of a sphere is determined by
V =43πr3 ).
a. g(t) = 8 − 0.6t
b. g(t) = 43πr3t
c. g(t) = 43π 8 − 0.6t 3( )
d. g(t) = 43π 8 − 0.6t( )3
e. g(r) = 43πr3
9. The distance, s (in feet), traveled by a car moving in a straight line is given by the function, ( ) ttts += 2 , where t is measured in seconds. Find the average velocity for the time period from t = 1 to t = 4.
a. sec5 ft b. sec6 ft c. sec9 ft d. sec10 ft e. sec11 ft
10. The graph below represents the number of points scored by a basketball team, P, as a
function of the number of minutes after the start of the game. Evaluate P(15) and explain its meaning.
a. P(15) = 10; Fifteen minutes after the start of the game the team had scored 10 points. b. P(15) = 10; Ten minutes after the start of the game the team had scored 15 points. c. P(15) = 20; Fifteen minutes after the start of the game the team had scored 20 points. d. P(15) = 20; Twenty minutes after the start of the game the team had scored 15 points. e. P(15) = 8; Fifteen minutes after the start of the game the team had scored 8 points.
11. A word processor determines the width of the body of text on a page for a margin
setting of x inches. If the page is originally 8.5 inches wide and has two equal size margins on each side, which formula determines the width of the body of the text?
a. b. c. d. e. 2x
2x − 8.5x − 8.58.5 − 2x8.5 − x
12. Use the graph of f to solve f (x) = −3 for x.
a. (−3, −2) b. −4 c. (−4, −3) d. −2 e. −3
13. The variables x and y in the table below are in
a linear relationship. What is the value of y when x is 9? a. 0 b. 7 c. 1 d. –1 e. –4 14. Suppose you determine the average speed of this racecar over the interval t = 1
to t = 4 sec. Which statement best describes the meaning of average speed in this context?
a. The sum of the initial and final speeds divided by 2 b. The speed that the racecar drove most of the time. c. The constant speed that the racecar traveled during the 3 seconds. d. The constant speed needed to travel the same distance as the racecar in the
same amount of time (3 seconds). e. The sum of the distances at t = 1,2,3,4 divided by 4
15. If S(m) represents the salary (per month), in hundreds of dollars, of an
employee after m months on the job, what would the function R(m) = S(m + 12) represent?
a. The salary of an employee after 12 months on the job. b. $12 more than the salary of someone who has worked for m months. c. An employee who has worked for m + 12 months. d. The salary of an employee after m + 12 months on the job. e. Not enough information.
x y 1 15 3 11 9 12 –7
Use the graphs of f and g to answer items 16 and 17. 16. Use the graphs of f and g to evaluate . a. –2 b. 1 c. 3 d. 4 e. Not defined 17. Evaluate . a. -4 b. -‐2 c. 0 d. 2 e. 4 18. Let S(m) represents the salary (per month), in hundreds of dollars, of an employee after
m months on the job. What does the solution to the equation S(m) = 23 represent? a. The salary of an employee after 23 months on the job. b. The monthly salary of the employee is 23. c. The number of months it would take for an employee to earn a salary of $2,300 per
month. d. The monthly salary of the employee is $2,300. e. The number of months it would take for an employee to earn a salary of $23 per
hour. 19. Given the function and , evaluate .
a. 10 b. 11 c. 20 d. 25 e. 36
20. Given that f(1) = 6 and f(5) = 14, what is the average rate of change of f on the interval from x = 1 to x = 5. a. 10 c. 8 c. 2 d. 4 e. 6
g f 2( )( )
f 2( ) − g 0( )
h x( ) = 3x −1 g x( ) = x2 g h 2( )( )
21. What is the domain of the following function: ( ) ?12
−+=
xxxf
a. x >1 b. 1≠x c. x > −2 and x ≠ 1 d. x > −2 e. All real numbers
22. At 5:00 am Maria leaves a gas station and drives on I-‐70 toward Denver at a constant rate of 40 miles per hour. At 5:30 am Robert leaves the same gas station and also drives on I-‐70 towards Denver at a constant rate of 60 miles per hour. At 6:00 am how far apart are Maria and Robert?
a. 0 miles b. 10 miles c. 20 miles d. 30 miles e. 40 miles
23. A baseball card increases in value according to the function, b(t) = 52 t + 100, where b
gives the value of the card in dollars and t is the time (in years) since the card was purchased. Which of the following describe what 2
5 conveys about the situation?
I. The card’s value increases by $5 every two years. II. Every year the card’s value is 2.5 times greater than the previous year.
II. The card’s value increases by 25 dollars every year. a. I only b. II only c. III only d. I and III only e. I, II and III
24. Two cars that are originally 216 miles apart start traveling toward one another at 9 am.
One car travels 62 miles per hour and the other travels 68 miles per hour, both with their cruise control on. Which formula represents d, the distance between the two cars t hours after 9 am.
a. d =62 68t t+
b. d = 216− 62t
c. d =68 62t t−
d. d = 216 (68 62 )t t− +
e. d = 216+ (68t + 62t)
25. Given that 5 10Q P= + , which of the following is correct?
a.
b.
c.
d.
A racecar travels around a circular track for 2083 seconds. Let d represent all possible values of the distance of the racecar (in meters) from the starting line, and let t represent all possible values of the number of seconds since the racecar left the starting line. The relationship between d and t is given by the formula f(t) = 0.6t2 + 2t where d = f(t). 26. What is the meaning of f(5) – f(3)?
f. The time elapsed as the car moved from the 3 m mark to the 5 m mark g. The distance traveled by the car from t = 3 to t = 5 seconds h. The increase in speed as the car moved from the 3 m mark to the 5 m mark i. The average speed of the car from t = 3 to t = 5 seconds e. None of the above
27. Evaluate f(5) – f(3).
a. 6.4 b. 11.4 c. 13.6 d. 25 e. None of the above
28. What is the average speed of the racecar as the number of seconds since it left the starting
line increases from 1 to 4 seconds?
a. 3 m/s b. 5 m/s c. 15 m/s d. 17.6 m/s e. None of the above
29. Which statement best describes the meaning of average speed in this context?
f. The sum of the distances at t = 1, 2, 3, 4 divided by 4 g. The speed that the racecar drove most of the time. h. The constant speed that the racecar driver traveled during the 3 seconds. i. The constant speed needed to travel the same distance as the given car in the same
amount of time (3 seconds). j. The sum of the initial and final speeds divided by 2
See open-‐ended questions on next page
P = 15Q +10
P = 15Q −10
P = 15Q − 2
P = 15Q +10
Open-‐Ended Section – Show all work and explain all answers thoroughly on this exam. Be sure to define variables where needed.
30. (12 pts) A large spool is used to hold rope that is wound around the spool. The more rope wound around the spool, the greater the combined weight of the spool and rope. The graph below shows that when 5 feet of rope is wound around the spool, the total weight of the spool and rope is 3.95 pounds. Note that the rope weighs 0.27 pounds per foot.
a. Suppose the number of feet of rope on the spool increases from the given point to 8.4 feet. What is the change in the number of feet of rope on the spool?
b. Represent this change on the graph above. Explain the thinking you used to represent this change.
c. What is the total weight of the spool and rope when there are 8.4 feet of rope on the spool? Explain how you determined your answer.
d. What is the weight of the spool without any rope? Explain how you determined this value and represent your reasoning on the graph above.
Total length of rope on the spool (in feet)2 4 6 8 12
Tota
l wei
ght o
f spo
ol a
nd ro
pe (i
n po
unds
)
1
2
3
4
5
(5, 3.95)
31. A 3 m x 1.5 m piece of plywood is being used to build an open-‐top toy chest. The chest is formed by making equal-‐sized square cutouts from two corners of the plywood at the ends of a 3-‐meter side. After these squares are discarded, three more cuts are made (at the dashed lines on the figure) and these pieces are “folded up” and secured to create three of the four vertical sides. When the open side is placed against a wall, the open top toy chest is formed. Neglect the thickness of the wood when answering these questions.
a. If the variable x represents the length of the sides of the square cutouts in meters, define a function f that expresses the volume of the toy chest (in cubic meters) as a function of the length of the side of the square cutouts, x.
b. What are the possible values that x can assume in the context of this problem?
c. As the length of the side of the square cutout increases from 0.5 meters to
0.75 meters, how does the volume of the toy chest change? (You do not need to calculate the answer. You may just write the expression that you would use to calculate the answer.)
32. The air temperature, T, in degrees Fahrenheit, is given in terms of the chirp rate, R, in chirps per minute, of a snowy tree cricket by the function f where 𝑇 = 𝑓 𝑅 and 𝑓(𝑅) = !
!𝑅 + 40.
a. Determine the function rule for the inverse function 𝑓!!
b. What does the inverse function represent in this situation?
Additional Skill Problems:
33. If f x( ) = 6x2 −1
x − 2 find f 0( ) .
34. If g x( ) = 4x − 5 , find g 2 + h( ) − g 2( )
h. Simplify your answer as far as possible.
35. If h x( ) = x2 − 3x and k x( ) = x +1 , find h k 8( )( ) .
36. If y = l x( ) = 4x + 72x
, find l−1 y( ) .
37. If R x( ) = 300x and C x( ) = 200 +175x , find the value of x where R x( ) = C x( ) .