Download Calculus Individual FAMAT State Convention 2012 For each

Calculus Individual
FAMAT State Convention 2012
For each question, answer “E) NOTA” means none of the given answers is correct.
1)
The curves (i), (ii), and (iii) in the graph below are the graphs of a continuous function 𝑓 and its first and second
derivatives. Which curve represents 𝑓, which is 𝑓′, and which is 𝑓′′?
A) (i) 𝑓 (ii) 𝑓′′ (iii) 𝑓′
B) (i) 𝑓′′ (ii) 𝑓′ (iii) 𝑓
2)
E) NOTA
𝑛 + 𝑒 2ℎ , ℎ ≥ 0
The function given by 𝐺(ℎ) = {
is differentiable at ℎ = 0. What is the value of 𝐺(𝑛 − 𝑚)?
4 + 𝑚ℎ, ℎ < 0
A) 2
3)
C) (i) 𝑓′′ (ii) 𝑓 (iii) 𝑓′
D) (i) 𝑓’ (ii) 𝑓′′ (iii) 𝑓
C) 2 + 𝑒
B) 4
D) 3 + 𝑒 2
E) NOTA
Which of the following statements are true?
I: The graphs of 𝑀(𝑥) = 𝑒 𝑥 and 𝐴(𝑥) = 𝑒 −𝑥 + 2012 meet at right angles.
II: If 𝑂(𝑥) = ln 𝑥, then 𝑂(𝑒 𝑛+1 ) − 𝑂(𝑒 𝑛 ) = 1 for any value of n.
III: If 𝑓(𝑥) = 𝑔(𝑥)𝑒 𝑥 , then the only zeros of 𝑓 are the zeros of 𝑔.
A) Statement III only
B) Statements I and III
4)
E) NOTA
The hypotenuse AB of the right triangle ABC remains constant at 5 feet as both legs are changing. One leg, AC
is decreasing at the rate of 2 feet per second. In order for the hypotenuse to remain 5 feet, the other leg BC is
increasing. What is the rate, in square feet per second, at which the area is changing when AC = 3?
A) -7/2
B) -7/4
5)
C) Statements II and III
D) Statements I, II, and III
C) -3/2
D) 25/4
E) NOTA
The graph of a function 𝑓 consists of three line segments (shown below).
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Calculus Individual
FAMAT State Convention 2012
3
Define ℎ(𝑥) = ∫𝑥 𝑓(𝑡)𝑑𝑡. Let 𝐼 = the number of inflection points of ℎ. Also, let 𝐷 = the length of the intervals
where ℎ is decreasing. What is 𝐷 − 2𝐼?
A) -2
6)
B) 0
𝑒𝑥
If 𝑚(𝑥) = ∫2
𝑑𝑡
√1+𝑡 4
C) 3
D) 5
, write (𝑚−1 )′ (0) in the completely reduced and rationalized form:
E) NOTA
𝑎√𝑏
,
𝑐
where 𝑎, 𝑏, 𝑐 are
positive integers and 𝑎 and 𝑐 are relatively prime. What is 𝑏 – 𝑐𝑎?
A) 9
7)
B) 10
C) 15
D) 16
E) NOTA
The function 𝑔 is defined and differentiable on the closed interval [−7, 5] and satisfies 𝑔(0) = 5. The graph of
𝑦 = 𝑔′ (𝑥) consists of a semicircle and three line segments, as shown in the figure below.
1
Now, the function ℎ is defined by ℎ(𝑥) = 𝑔(𝑥) − 2 𝑥 2 . What is the value of the y-coordinate of the relative
maximum of ℎ (rounded to the nearest integer) in the interval [−7, 5]?
A) 2
B) 6
8)
C) 8
D) 9
E) NOTA
The closed interval [𝑎, 𝑏] is partitioned into 𝑛 equal subintervals, each of width ∆𝑥, by the numbers
𝑥0 , 𝑥1 , … , 𝑥𝑛 , where 𝑎 = 𝑥0 < 𝑥1 < 𝑥2 < ⋯ < 𝑥𝑛 = 𝑏. What is lim ∑𝑛𝑖=1 √𝑥𝑖 ∆𝑥?
𝑛→∞
1
A)
B)
1
3
𝑏 2 − 𝑎2
1
2
1
2
2 (𝑏 − 𝑎 )
3
C)
𝑏 2 − 𝑎2
D)
3
2
3
2
E) NOTA
3
2
(𝑏 − 𝑎 )
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Calculus Individual
FAMAT State Convention 2012
(9)
2𝑧 3
2
9) Let 𝐷(𝑧) = 𝑒
+ 𝑧𝑐𝑜𝑠(3𝑧 ). What is 𝐷 (0)? That is, what is the ninth derivative of D evaluated at 0?
A) 9!
31
8
B) 9!
85
24
C) 9!
49
24
D) 9!
113
24
E) NOTA
2𝑥 5 −14𝑥 3
10) Consider the following function: 𝑞(𝑥) = 5 4 3
. If q(x) has a horizontal asymptote at 𝑦 = 𝑀 and
𝑥 +4𝑥 −7𝑥 −28𝑥 2
we let 𝐴 and 𝑂 represent the number of vertical and oblique asymptotes, respectively, what is 𝑀 + 𝐴 + 𝑂?
A) 6
B) 5
C) 3
D) 2
E) NOTA
𝑥
11) What is the range of the inverse of 𝐵(𝑥) if 𝐵(𝑥) = 𝑥−3?
A) (−∞, ∞)
B) All real numbers except x = 3
C) All real numbers except x = 1
D) (−∞, 1) ∪ (3, ∞)
E) NOTA
12) Sabrina “I pity the non-calculus student” T. likes to bake cookies. She keeps track of her daily outputs, and has
determined that the main determinant of output is the pounds of dough she bought the night before. She models
the relationship between these variables, and has derived a function 𝑇(𝑑) that is differentiable on the interval
[6, 15]. One day, Sabrina T. accidently dropped coffee on her computer and lost the model. However, she
remembers that 𝑇 ′ (𝑑) ≤ 10 and that when she bakes 6 pounds of dough, she always produces 12 cookies. What
is the largest possible value for 𝑇(15)?
A) 138
B) 129
C) 102
D) 36
E) NOTA
13) Suppose 𝑔(𝑥) and ℎ(𝑥) are such that lim 𝑔(𝑥) = ∞ and lim ℎ(𝑥) = 0, where 𝑎 is a real number. Which of the
𝑥→𝑎
𝑥→𝑎
following statements are FALSE?
I. lim [𝑔(𝑥)ℎ(𝑥)] is never equal to 0.
𝑥→𝑎
II. lim [𝑔(𝑥)ℎ(𝑥)] may be equal to a positive value.
𝑥→𝑎
A) Statement I only
B) Statement II only
C) Both Statements I and II
D) Neither Statement I nor II
E) NOTA
14) There are 600 PHUHS students in line for the thrilling Sheikra roller coaster when the ride begins operation in
the morning. Once it begins, the ride accepts passengers (only PHU students!) until Busch Gardens closes eight
hours later. While there is a line, students move onto the ride at a rate of 800 students per hour. The graph
below shows the rate, 𝑟(𝑡), at which students arrive at the ride throughout the day. Time 𝑡 is measured in hours
from the time Sheikra begins operation.
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Calculus Individual
FAMAT State Convention 2012
Let 𝑌 = the number of people that are in line at the time when the line for Sheikra is the longest. What is the
sum of the digits of 𝑌?
A) 5
B) 7
C) 8
D) 11
E) NOTA
15) Sarah K. loves differentiation so much that she even dreams of it. Unfortunately, she had a nightmare about the
function 𝑦(𝑥) = 𝑥 𝑙𝑛𝑥 because she could not differentiate it. She needs your help! What is 𝑦 ′ (𝑒)?
A) 1
B) 𝑒
D) 𝑒 2
C) 2𝑒
E) NOTA
16) What is the area of the region bounded by 𝐻(𝑥) = 𝑒 √𝑥 , the x-axis, and the lines 𝑥 = 0 and 𝑥 = 4?
A) 2𝑒 2
B) 4𝑒 2
C) 2(𝑒 2 − 1)
D) 4(𝑒 2 − 1)
E) NOTA
17) Dr. Doom and Mr. Cody are in an epic 𝜀 − 𝛿 battle of limits. Mr. Cody is trying to counter each of Dr. Doom’s
𝜀 with an appropriate 𝛿 so that Mr. Cody can prove the limit exists. The limit of interest is lim (4𝑥 − 1). If Dr.
𝑥→1
Doom uses 𝜖 = 0.001, which of the following 𝛿 can Mr. Cody use to counter Dr. Doom? Assume that 0 <
|𝑥 − 1| < 𝛿.
A) 1
B) 0.004
C) 0.001
D) 0.00025
E) NOTA
18) Greg is on a boat! He likes to drive his boat on Lake Tarpon as long as the alligators are hibernating. Greg’s
speed 𝑣 (in kilometers per hour) at certain times during the day is given in the table below.
Time
Speed
11AM
32
11:30AM 12PM
30
16
12:30PM
22
1PM
20
1:30PM
24
2PM
26
2:30PM
30
Because Greg is a nerd, he wants to approximate the total distance he traveled from 11:30AM to 2PM using a
left Riemann sum. Which of the following best approximates the distance Greg traveled in this time interval?
A) 56
B) 66
C) 78
D) 85
E) NOTA
19) Which of the following statements is true?
I: If 𝐵′ (𝑥) > 0 for all real numbers 𝑥, then 𝐵(𝑥) increases without bound.
II: If 𝑋 ′′ (𝑦) < 0 for all real numbers 𝑦, then 𝑋(𝑦) decreases without bound.
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Calculus Individual
FAMAT State Convention 2012
A) Statement I only
B) Statement II only
C) Both statements I and II
D) Neither statement I nor II
E) NOTA
20) Mrs. Lindar has discovered a very special function, which she named the Narwal function (denoted by 𝑁𝑎𝑟(𝑥)).
The Narwal function is special because any tangent to the graph of 𝑁𝑎𝑟(𝑥) at the point (𝑥0 , 𝑦0 ) intersects the yaxis at (𝑥0 + 2, 0). In addition, the function intersects the line 𝑓(𝑥) = 𝑥 at 𝑥 = 2. What is 𝑁𝑎𝑟(4)?
A)
2
𝑒2
B)
2
𝑒
D) 2𝑒 2
C) 2𝑒
E) NOTA
21) What is the volume of the solid formed by revolving the region bounded by the graphs of 𝑦 = 𝑥 3 + 𝑥 + 1, 𝑦 =
1, and 𝑥 = 1 about the line 𝑥 = 2?
A)
89𝜋
30
B)
41𝜋
30
C)
49𝜋
30
D)
29𝜋
15
E) NOTA
22) Which of the following statements are FALSE?
I: The zeros of 𝑓(𝑥) = 𝑝(𝑥)/𝑞(𝑥) coincide with the zeros of 𝑝(𝑥).
II: If 𝑓(𝑥) is a cubic polynomial such that 𝑓 ′ (𝑥) is never zero, then any initial
guess will force Newton’s Method to converge to the zero of 𝑓.
III: If the coefficients of a polynomial function are all positive, then the
polynomial has no positive roots.
A) Statement I
B) Statements I and II
C) Statements II and III
D) Statements I, II, and III
E) NOTA
1
𝑛! 𝑛
𝑛
23) Evaluate: lim ln ( 𝑛 )
𝑛→∞
A) Does not exist
B) −𝑒
C) -1
D) 0
E) NOTA
24) Let ℓ be the line tangent to the graph of 𝑦 = 𝑥 𝑛 at the point (1,1), where 𝑛 > 1. In addition, let 𝑇 be the
triangular region bounded by ℓ, the x-axis, and the line 𝑥 = 1. Let 𝑆 be the region bounded by the graph of 𝑦 =
𝑥 𝑛 , the line 𝓁, and the x-axis.
What is the value of 𝑛 that will maximize the area of 𝑆?
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Calculus Individual
A) √3
FAMAT State Convention 2012
B)
1
√2
+ 2
2
C) 1 + √2
D) 2√3
E) NOTA
𝑥
25) A certain function 𝑔(𝑥) has the property that ∫0 𝑔(𝑡)𝑑𝑡 = 𝑒 𝑥 𝑐𝑜𝑠𝑥 + 𝐾 . What is the value of 𝑔(0) + 𝐾?
A) -1
B) 0
1
∞
26) What is the value of ∑𝑛=1 4𝑛2 −1?
A) -1/2
B) 1/2
C) 1
D) 2
E) NOTA
C) 1/4
D) 1/8
E) NOTA
27) Dani Vandi is traveling on a straight road. For 0 ≤ 𝑡 ≤ 24 seconds, the car’s velocity 𝑣(𝑡), in meters per
second, is modeled by the piecewise-linear function defined by the graph below.
Which of the following statements is FALSE?
I:
Dani travels 320 meters in 24 seconds.
II:
Dani’s acceleration at 𝑡 = 4 seconds is 5 m/sec2.
III: Dani is guaranteed to travel the average rate of change of 𝑣 between 8 and 20 seconds at some time
between 𝑡 = 8 and 𝑡 = 20 seconds.
A) Statement I only
B) Statement III only
C) Statement I and II
D) Statement II and III
E) NOTA
28) Ms. Fish is in a serious dilemma. She promised her good friend Mr. Mac that she would determine the
1
approximate value of ∫0 √1 + 𝑥 2 𝑑𝑥 using the Trapezoidal Rule. Unfortunately, she forgot the number of
intervals (called 𝑛) that she needed to ensure the error of her approximation was less than 0.01. Help Ms. Fish!
What is the smallest 𝑛 (rounded to the nearest integer) that she can use?
A) 2
B) 3
C) 8
D) 9
E) NOTA
29) If 𝑄(𝑥) = 𝑥 4 − 5𝑥 2 + 4, 0 ≤ 𝑥 ≤ 2, then which of the following are true?
I:
The absolute maximum is 4.
II:
An endpoint local minimum occurs at 𝑥 = 0.
III:
5
The critical numbers are 𝑥 = 0, ±√2
A) Statement I only
B) Statement I and II
C) Statement I and III
D) Statement II and III
E) NOTA
30) What is the slope of a line that is perpendicular to the tangent line of the curve defined implicitly by 𝑥 2 (𝑥 2 +
√2 √2
)?
2
𝑦 2 ) = 𝑦 2 at the point ( 2 ,
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Calculus Individual
A) 3
FAMAT State Convention 2012
B) 3/2
C) -1/3
D) -2/3
E) NOTA
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