Mathematics 2122-005
Calculus for Life Sciences II
Fall 2015
Final
Instructor: Dr. Alexandra Shlapentokh
(1) What is the antiderivative of cos x?
(a) − sin x
(b) tan x
(c) sin x
(d) tan x
(e) None of the above
(2) What is the antiderivative of 10x4 ?
(a) x6
(b) 2x5
(c) 2x4
(d) 21 x5
(e) NoneRof the above
(3) Compute (ex + 2x + x2 )dx.
(a) ex + x2 − ln(x) + C, where C is a constant
(b) ex + x2 − ln(|x|)
(c) ex + x2 + ln(x2 ) + C, where C is a constant
(d) ex + x2 + ln(|x|)
(e) None
Z of the above
Z
(4) What is g(x)dx if 3g(x)dx = 2 sin x + C, where C is an arbitrary constant?
(a) 23 sin(x) + C, where C is an arbitrary constant.
(b) sin(x/2) + C, where C is an arbitrary constant.
(c) 13 sin(x) + C, where C is an arbitrary constant.
(d) 32 sin(x) + C, where C is an arbitrary constant.
(e) NoneRof the above
(5) Compute e3x dx.
(a) 3e3x + C, where C is an arbitrary constant.
(b) 13 e3x + C, where C is an arbitrary constant.
(c) 13 ex/3 + C, where C is an arbitrary constant.
(d) 23 ex/3 + C, where C is an arbitrary constant.
(e)
R None of the above
R
(6) If g(x)dx = sin(x) + C, where C is an arbitrary constant, then what is g(3x)dx?
(a) sin(3x) + C, where C is an arbitrary constant.
(b) 3 sin(x/3) + C, where C is an arbitrary constant.
(c) 31 sin(3x) + C, where C is an arbitrary constant.
(d) 3 sin(x/3) + 3C, where C is an arbitrary constant.
(e)
R None of the above
R
(7) If g(x)dx = sin(x)+C, where C is an arbitrary constant, then what is g(x/3)dx?
1
(a) sin(2x) + C, where C is an arbitrary constant.
(b) 3 sin(x/3) + C, where C is an arbitrary constant.
(c) 13 sin(3x) + C, where C is an arbitrary constant.
(d) 13 sin(3x) + 13 C, where C is an arbitrary constant.
(e) NoneZof the above
π/2
−2 csc2 (x) + 2dx
(8) Compute
π/4
√
(a) 3
(b) does not exist
(c) 1
(d) −2 + π/2
(e) None of the above
(9) The area bounded by the curves y = −2x2 + 4 and y = 2x + 4.
5
(a)
6
7
(b)
3
1
(c)
3
11
(d)
6
None of the above
Z(e)
2
2
(10)
6xex − 12dx
0
(a) 9e4 + 2
9
(b)
2
(c) 3e4 − 27
1
9
(d) e4 +
2
2
(e)
None
of
the above
Z π/2
12 cot(x) + 12dx
(11)
π/4
(a) 2 + 3π/4
1
π
(b) ln(2) −
2
4
(c) 6 ln(2) + 3π
1
1
π
(d) ln( ) +
2
2
2
None of the above
Z(e)
4
x−1
(12)
10 2
− 10dx
x − 2x
3
(a) 5 ln( 83 ) − 10
(b) 12 ln( 38 ) − 2
(c) ln( 38 ) + 1
(d) 12 ln(5)
(e) None of the above
2
√
6 x + 1 − 12dx
1
√
√
(a) −
27
+
√
√ 7−6
(b) √27 + 6√− 5
(c) 4 √27 − 8 √2 − 12
(d) 12 3 27 − 21 3 8 + 1
None of the above
Z(e)
2
x+2
(14)
6 2
+ 24dx
x + 4x
1
(a) 24 + 3 ln 2.4
(b) 12 + ln 2.1
(c) 12 + ln 2.2
(d) 1 + 21 ln 2.3
Z(e) None of the above
Z
2
(13)
π/2
3 cos2 (x) sin(x) − 5dx
(15)
0
5π
(a) 1 −
2
π
(b)
2
π
(c) −2 +
2
π
(d) 3 −
2
None of the above
Z(e)
1
−8 ln3 (x)
− 6dx
(16)
x
e
(a) 2 + 3e
(b) 3 − e
(c) 6e − 4
(d) 4 + 3e
(e) None of the above
In Problems 17 and 18 assume that F 0 (x) = f (x) and C is a constant. Determine
which
statements below are always true under the given assumptions.
Z
(17)
6 cos x − 6f (− cos x) sin xdx
(a)
(b)
(c)
(d)
Z(e)
= F (cos x) + sin x + C
= F (sin x) − cos x + C
= −6F (− cos x) + 6 sin x + C
does not exist
None of the above
4
(18)
4f (e−x )e−x − dx
x
(a) = F (ln x) + ln x + C
(b) = F (e−x ) + ln(x) + C
(c) = −4F (e−x ) − 4 ln(|x|) + C
3
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(d) does not exist
(e) None of the above
Find the
√ area between y = 4 sin x + 2 and y = 4 cos x + 2 for x ∈ [0, π/4].
(a) 4√2 − 4
(b) 2 √2 − 1
(c) −
√ 2−1
(d) 2 + 1
(e) None of the above
Find the
√ area between y = 8 sin x − 3 and y = 8 cos x − 3 for x ∈ [0, π/2].
(a) 2 √
2−1
(b) −2√ 2 − 1
(c) 16√ 2 − 16
(d) 8 2 + 8
(e) None of the above
Compute ∂f /∂x for f (x, y) = 8y 2 + 4.
(a) x
(b) y
(c) ∂f /∂x does not exist.
(d) 0
(e) None of the above
Compute ∂f /∂y for f (x, y) = 6y 2 + x − 18.
(a) x
(b) 0
(c) ∂f /∂y does not exist.
(d) 12y
(e) None of the above
Compute ∂ 2 f /∂x∂y for f (x, y) = 2y 2 + 12x + 81.
(a) (x − y)2
(b) x
(c) 0
(d) This function does not have this second order partial derivative.
(e) None of the above
Suppose an antiderivative of y = f (x) is esin x + 30x. Then f (x)
(a) cannot be determined.
1
(b) is
.
csc x
(c) is esin x .
(d) is esin x cos x + 30.
(e) None of the above
Suppose f 0 (x) = 80t4 − 16, f (0) = 0. Determine f (1).
(a) -1
(b) 0
(c) 1
(d) 2
(e) None of the above
4
Z
(26) If a ≤ b ≤ c are real numbers and
b
Z
c
f (x)dx = −4, then
f (x)dx = 6,
b
a
Z
is equal to
(a) 2
(b) -1
(c) 0
(d) 2
(e) None of the above
(27) Consider a function y = f (x) with the following graph.
c
f (x)dx
a
20
10
–3
–2
–1 0
1
x
2
3
–10
–20
The area bounded by the graph of y = f (x), lines x = −2.2, x = 1, and y = 0 is
equalZto
1
(a)
f (x)dx
Z−2.2
0
(b)
1
Z
f 2 (x)dx
f (x)dx +
Z−2.2
0
Z0 1
f (x)dx −
(c)
−2.2
Z 0
(d) −
f (x)dx
0Z
f (x)dx +
−2.2
1
f (x)dx
0
(e) None of the above
(28) Suppose a particle is moving along a straight line with the acceleration a(t) = 12t
as a function of time measured in hours. Let v(t) be the velocity as a function of
time and let s(t) be the position as the function of time, also measured in hours. If
v(0) = 4 and s(0) = 0, then what is s(1)?
(a) 10/3
(b) 6
(c) -5/3
(d) -3
(e) None of the above
(29) Suppose P 0 (t) = 54e2t for t ∈ [0, 4], where P (t) is the number of individuals in a
certain population at time t. What is the increase in the population between t = 1
and t = 4.
(a) 9e4 + 54e
(b) 27e8 − 27e2
(c) 27e
(d) 27e3 − 3e
5
(e) None of the above
(30) Let y(t) be a solution to 6y 0 = 24 + 6y, with y(0) = 1. What is y(4)?
(a) 4 − 2e4
(b) −4 + 5e4
(c) 4 − 5e4
(d) 3 + 3e4
(e) None of the above
(31) If y = Ce3x , where C 6= 0 is a constant, is a general solution to a differential
equation, then what particular solution is obtained from the initial value y(0) = 11?
(a) y = 5x
(b) y = 5e3x
(c) There is not enough information to answer this question.
(d) y = 11e3x
(e) None of the above
(32) What is the general solution to the equation y 0 = ky?
(a) y = ekx
(b) y = 2ekx
(c) y = Cekx , where C is a constant.
(d) This equation does not have a solution.
(e) None of the above
(33) What√is y(1) if 3y 0 + 6xy 2 = 0 and y(0) = 1?
(a) 2
(b) 2
(c) 12
(d) e − 1
(e) None of the above
(34) What is a general solution to the equation y 0 = f (x), where f (x) is a continuous
function?Z
(a) y = f (x)dx
(b) y = f 0 (x)
(c) y = f 0 (x)
Z − C, where C is a constant
(d) y = C
f (x)dx, where C is a constant.
(e) None of the above
(35) What is y(π/2), if 4y 0 = 32 cos t and y(0) = 0?
(a) 1
(b) 8
(c) 3
(d) There are not enough data to answer this question.
(e) None of the above
(36) What is y(Π) if 4y 00 = 32 cos x, y 0 (0) = 0, y(0) = 1.
(a) 2Π − 1
(b) 2
(c) 1
(d) −2Π − 1
6
(e) None of the above
(37) Which of the following statements is true?
(a) y = 2 cos x is a solution to the differential equation y 00 = −y.
(b) y = e2x is a solution to the differential equation y 000 = 3ex .
(c) y = sin x is not a solution to the differential equation y 00 = −y.
(d) y = 2x2 is a solution y 000 = 3x
(e) None of the above
(38) Consider a differential equation y 0 + p(x)y = q(x) and determine which of the
statements below is true for any p(x), q(x) continuous everywhere.
(a) [G(x)y]0 = q(x)G(x), where G(x) = eF (x) and F (x) is any antiderivative of
p(x).
(b) G(x)0 = q(x)G(x), where G(x) = eF (x) and F (x) is any antiderivative of p(x).
(c) y 0 = q(x)G(x), where G(x) = eF (x) and F (x) is any antiderivative of p(x).
(d) [G(x)y]0 = G(x), where G(x) = eF (x) and F (x) is any antiderivative of p(x).
(e) None of the above
(39) Consider the following system:

 x + y + 2z = 9
x − y + 3z = 8

x + y − 2z = −3
Which of the following statements about the system is true?
(a) All the solutions are of the form (a, a, a), where a is a real number.
(b) All the solutions are of the form (a, b, c), where a < b < c and a, b, c are real
numbers.
(c) All the solutions are of the form (a, b, b), where a > b and a, b are real numbers.
(d) All the solutions are of the form (b, b, a), where a > b and a, b are real numbers.
(e) None of the above
(40) What y(−2) if 2y 0 + x8 y − 4x = 0 and y(2) = 2?
(a) 3
(b) −2
(c) 2
(d) 0
(e) None of the above
(41) Suppose a tank contains 50 gal of water and 2 pound of a certain chemical. A
solution of the same chemical containing 2 pound per gallon is being poured into
the tank at the rate of 1 gallon an hour while the tank is being drained at the same
rate. Let A(t) be the amount of the chemical in the tank at time t. Assume the
distribution of the chemical in the tank is uniform at any moment of time. Write
down the initial value problem to determine A(t).
A
(a) A0 = 1 + 100
, A(0) = 1
A
0
(b) A = −1 − 100 , A(0) = 4
A
(c) A0 = 2 − 50
, A(0) = 2
A
0
(d) A = 2 − 50 , A(0) = 3
(e) None of the above
7
F IGURE 1.
10
f (x)
5
−2
−1
1
2
g(x)
−5
(42) Given a tank as in Question 41, how many pounds of the chemical are there in the
tank after 100 hours?
(a) 100 + 99e−2
(b) 100 − 98e−2
(c) 100 − 99e−1
(d) 600 − 596e−1
(e) None of the above
(43) Given a tank as in Question 41, if the process continues indefinitely, how many
pounds of the chemical will there be in a tank in the long run?
(a) 800
(b) 700
(c) 100
(d) 500
(e) None of the above
(44) If 6y 0 − 4x
= 0, y(0) = 1, then what is y(1)?
y2
√
3
(a) −
√ 2
(b) √2
3
3
(c) √
(d) 3 2
(e) None of the above
(45) Which of the following statements is true?
(a) A linear system always has a unique solution.
(b) A linear system always has at most one solution.
(c) A linear system can have exactly 4 solutions.
(d) A linear system can have exactly 3 solutions.
(e) None of the above
Z
(46) Compute the shaded area in Figure 1 from the data given below.
Z 1
7,
g(x) = −4.
−1
(a) 11
(b) 4
(c) 5
8
1
f (x) =
−1
F IGURE 2.
10
f (x)
5
g(x)
−2
−1
1
2
(d) 6
(e) None of the above
Z
(47) Compute the shaded area in Figure 2 from the data given below.
Z 1
g(x) = 3.
8,
1
f (x) =
−1
−1
(a) 0
(b) 5
(c) 2
(d) 11
(e) None of the above
(48) Let A and B be 4 × 4 matrices and assume all entries of B are equal to 0. Let
C = AB and determine which of the statements below is true.
(a) C is not defined because you cannot multiply two matrices of the same size.
(b) All entries of C are equal to 0.
(c) The entries of C will depend on A. So for some A, some entries of C are not
equal to 0.
(d) At least one entry of C is equal to 1.
(e) None of the above
(49) Let A and B be 3 × 3 matrices and assume all entries of B on the main diagonal
are 1’s and off the main diagonal are zero. Let C = AB and determine which of
the statements below is true.
(a) Sometimes C = A but not always.
(b) All entries of C are zero.
(c) C = A
(d) The product AB is not defined.
(e) None of the above
(50) Let A and B be 2 × 2 matrices and assume all entries of A and B on the main
diagonal are 1’s and off the main diagonal are zero. Let C = A + B and determine
which of the statements below is true.
(a) all entries of C on the main diagonal are 2’s and off the main diagonal are
zero.
9
(b)
(c)
(d)
(e)
all entries of C on the main diagonal are 0’s and off the main diagonal are 1’s.
Not enough information is available to draw any conclusions about C.
The sum A + B is not defined.
NoneZof the above
1
[(200x)(x2 + 1)99 + 1]dx
(51) Compute
(a)
(b)
(c)
(d)
(e)
0
2100 + 2
2100
2100 − 1
2100 + 1
None of the above
10
Key:
1c, 2b, 3c, 4a, 5b, 6c, 7b, 8d, 9c, 10c, 11c, 12a, 13c, 14a, 15a, 16c, 17c, 18c, 19a, 20c,
21d, 22d, 23c, 24d, 25b, 26a, 26d, 27d, 28b, 29b, 30b, 31d, 32c, 33c, 34a, 35b, 36e, 37a,
38a, 39b, 40c, 41c, 42b, 43c, 44d, 45e, 46a, 47b, 48b, 49c, 50a, 51b.
11