HUNTER COLLEGE
MAT 155
Sample Final
May 2019
Print Name:
Instructor: H. Kone, Ph.D.
Directions: You have 110 minutes (1 hour 50 mins) to answer the following questions.
You do not need to show any work for multiple choice questions which worth
3 points each (question 1 to 18) but you must show all your work as neatly
and clearly as possible and indicate the final answer clearly for the free answers problems which worth 15 points each (question 19 to 21) .
You may use a non-graphing scientific calculator.
You can do this!
Problem Possible
Possible
MC
18
Total
54
Correct answers
3×
=
MC
54
FA 1
15
FA 2
15
FA 3
15
Bonus
1
Total
100
Points
1
5
(1) (Chapter 5). The formula C = (F − 32) where F ≥ −459.67, expresses the Cel9
sius temperature C as a function of the Fahrenheit temperature n. Then the inverse
function is
5
(A) F = (C − 32)
9
9
(B) F = C − 32
5
9
(C) F = C + 32
5
(D) F = 32 C −
9
5
(E) None of the above
(2) (Chapter 5). If $3000 is invested at 5% interest, then the value of the investment
at the end of 5 years if the interest is compounded continuously is
(A) $3000.00
(B) $3828.84
(C) $3840.25
(D) $3852.08
(E) None of the above
ex − e−x − 2x
=
x−→0
x − sin x
(3) (Chapter 5). lim
(A) 0
(B) 1
(C) 2
(D) Does not exist
(E) None of the above
(4) (Chapter 5). If f (x) =
√
3 − e2x , then the domain of f is
(A) (−∞, 3)
√
(B) (−∞, ln 3)
(C) (3, ∞)
(D) All real numbers
(E) None of the above
(5) (Chapter 5). If f (x) = ln(x2 − 9), then
(A) f 0 (x) =
x2
x
−9
(B) f 0 (x) =
x2
x2 − 9
(C) f 0 (x) =
−9
x2 − 9
(D) f 0 (x) =
2x
x2 − 9
(E) None of the above
Z
(6) (Chapter 6). The integral
arccos(x)dx =
0
(A) 0
(B) 1
(C) (π − 2)
(D) π
(E) None of the above
1
Z
(7) (Chapter 6). The integral
x2 ex dx =
(A) ex (2 + 2x + x2 )
(B) ex (2 − 2x + x2 ) + C
(C) ex (2 + 2x − x2 )+C
(D) −ex (2 + 2x + x2 ) + C
(E) None of the above
Z
π/2
cos4 x sin xdx =
(8) (Chapter 6). The integral
0
(A) 1
(B)
1
3
(C)
1
5
(D)
1
7
(E) None of the above
Z
(9) (Chapter 6). The integral
(A) ln
x2
x+3
+C
x+2
(B) 2 ln |x + 2| − ln |x − 3| + C
(C) ln
x+2
+C
x+3
(D) ln
|x − 3|
+C
(x − 2)
(E) None of the above
dx
=
+ 5x + 6
(10) (Chapter 6). If f (x) =
n = 10 on [1, 2] is
2
, then error bound of its midpoint approximation for
x
(A) 0
1
600
x
(C)
1200
1
(D)
1200
(B)
(E) None of the above
(11) (Chapter 7). A group of engineers is building a parabolic satellite dish whose shape
will be formed by rotating the curve y = ax2 about the y-axis. If the dish is to have
a 10-ft diameter and a maximum depth of 2 ft, find the value of a and the surface
area of the dish.
√
25π 2
(A) a = 2/5 and the surface area is
12
√
125π 2
(B) a = 2/25 and the surface area is
12
√
625π 2
(C) a = 5 and the surface area is
3
√
625π 2
(D) a = 2/25 and the surface area is
12
(E) None of the above
(12) (Chapter 8). Assuming that the pattern of the first few terms continues, the general
term of the sequence
4 8 16
−3, 2, − , , − , . . . is
3 9 27
2n
(a) an = (−1)n n−1 , n ≥ 0
3
(b) an = (−1)n
2n
, n≥0
3n−1
(c) an = (−1)n−1
2n
, n≥0
3n−1
(d) an = (−1)n−1
2n−1
, n≥0
3n
(e) None of the above
(13) (Chapter 8). The limit of the sequence
r q
(
)
q
√
√
√
2, 2 2, 2 2 2, . . .
is
(a) 0
(b) 1
(c)
√
2
(d) 2
(e) None of the above
(14) (Chapter 8). The values of x for which the series
∞
X
(x − 2)n
n=0
3n
converges and its
sum for those values are
(a) 0 < x < 1 and the sum is
3
5−x
(b) 0 < x < 5 and the sum is
3
1−x
(c) −1 < x < 5 and the sum is
3
5−x
(d) −1 < x < 5 and the sum is
1
1−x
(e) None of the above
(15) (Chapter 8). Determine whether the series
∞
X
n=1
If it is convergent, find its sum.
(a) This series diverges
(b) It converges and its sum is 0
(c) It converges and its sum is 1
(d) It converges and its sum is 2
(e) None of the above
1
is convergent or divergent.
n(n + 1)
(16) (Chapter 8). The series
∞
X
n=1
5
4n3
+
3n2
(a) Diverges
(b) Converges
(c) Converges to 0
(d) No enough information
(e) None of the above
(17) (Chapter 8). The series
∞
X
1 + 2n
n=0
(a) Diverges
(b) Converges
(c) Converges to
9
2
(d) No enough information
(e) None of the above
3n
is
+ 2n + 1
is
(18) (You must show all your work, chapter 5). A cup of hot chocolate has temperature
80◦ C in a room kept at 20◦ C . After half an hour the hot chocolate cools to 60◦ C.
You may use the Newton’s cooling law that gives you the temperature T at time t,
T (t):
T (t) = (Tinitial − Tsurroundings )e−kt + Tsurroundings .
(a) What is the constant of cooling k?
(b) What is the temperature of the chocolate after another half hour?
(c) When will the chocolate have cooled to 40◦ C?
(19) (You must show all your work, chapter 6). A tank contains 100 liters of water that
1
leaks at a rate r(t) =
liters per hour.
1 + t2
(a) Find the rate of leak at time t = 0.
(b) Find the total amount of water leaked during the first hour.
(c) What is the total amount of water that will leak if the leaking continues for
ever (i.e. as time t goes to infinity)?
(20) (You must show all your work, chapter 8). If $1000 is invested at 6% interest, compounded annually, then after years the investment is worth an = 1000(1.06)n dollars.
(a) Find the first five terms of the sequence {an }.
(b) Is the sequence convergent or divergent? Explain.
(21) (You must show all your work, chapter 8). A patient takes 150mg of a drug at the
same time every day. Just before each tablet is taken, 5% of the drug remains in
the body.
(a) What quantity of the drug is in the body after the third tablet? After the n-th
tablet?
(b) What quantity of the drug remains in the body in the long run?