Mathematics 2122-003
Calculus for Life Sciences II
Fall 13
Study Guide #2
Instructor: Dr. Alexandra Shlapentokh
(1) Suppose the velocity of a particle moving along a straight line is v(t) = t2 +1. What
is the distance traveled by the particle between t = 1 and t = 3?
(2) Suppose the rate of growth of a certain population between t = 1 hour and t = 3
hours is equal to t2 million of individuals per hour. What will be the increase in the
population during this time period?
(3) Suppose the sales were increasing between 2000 and 2009 at the rate of t2 +1−sin t
units per year, where t is the number of years since 2000. What was the the total
volume of sales between 2000 and 2009?
(4) Compute
definite integrals.
Z 4 the following
1 + 3x + 3x2
(a)
dx
5x
2
Z 6√
(b)
2xdx
Z5 9
1
(c)
dx
x
8
Z π/2
(d)
sin xdx
0
Z π/2
cos xdx
(e)
0
(5) Find the following areas.
(a) The area under y = x1 and above y = 0 between x = 1 and x = 3.
(b) The area bounded by y = sin x and y = 0 between x = −π and x = π.
(c) Find the area bounded by y = x2 − 9 and the x-axis between x = −3 and
x = 3.
(6) Find the average value of the following functions:
1
(a) y = , x ∈ [1, 10]
x
(b) y = sin x, x ∈ [−π, π]
(c) y = sin 2x, x ∈ [−π, π]
(d) y = sin x, x ∈ [0, π]
(e) y = sin 2x, x ∈ [−π, π]
(f) y = cos x, x ∈ [−π, π]
(g) y = cos x, x ∈ [0, π]
(7) What are the twoZversions of the Fundamental Theorem of Calculus?
x
et sin tdt. What is F 0 (x)?
(8) Suppose F (x) =
1
(9) What are the antiderivatives of the following functions?
1
= sin2 x cos x
= cos2 x sin x
3
= x2 ex
= sin(3x + 1)
ln2 (x)
(e) y =
x
(f) y = tan x
(g) y = cotZx
x
et
(10) Let F (x) =
dt. What is F 0 (x)?
2+1
t
0
(11) Compute
the
following definite integrals.
Z 6
√
2x + 1dx
(a)
Z5 100
x+1
(b)
dx
x2 + 2x
10
Z 1
2
ex
dx
(c)
1 ln x + 1
(12) Find the following areas.
(a) The area bounded
by the curves y = −x2 + 9 and y = x2 − 9.
Z 3
2
2
2
Solution:
(−x2 +9)−(x2 −9)dx = 18x− x3 |3−3 = 54− ·27+54− ·27 = 72.
3
3
3
−3
(b) The area bounded by sin x, cos x between x = 0 and x = π/4. Solution:
Z π/4
√
√
√
π/4
cos x − sin xdx = sin x + cos x|0 = 1/ 2 + 1/ 2 − 1 = 2 − 1
(a)
(b)
(c)
(d)
y
y
y
y
0
(c) The area bounded by sin x, cos x between x = 0 and x = π/2.
(d) Area bounded by the curves y = x2 + 2 and y = x + 2. R
1
(13) Let f (x), g(x) be two continuous functions on [−1, 1] with −1 f (x)dx = 7 and
R1
g(x)dx = 3 with f (x) > g(x) for all x ∈ [−1, 1]. Find the area bounded by the
−1
graphs ofZf (x) and g(x) between
Z x = −1 andZx = 1.
1
1
1
g(x)dx = 7 − 3 = 4
R1
(14) Let f (x), g(x) be two continuous functions on [−1, 1] with −1 f (x)dx = 7. Suppose
the area
Z bounded by the graphs of f (x) and g(x) between x = −1 and x = 1 is 5.
f (x)dx −
(f (x) − g(x))dx =
Solution
−1
−1
−1
1
Find
g(x)dx.
−1
Solution: there is not enough information to answer the question. One need to
know the relative positions of f (x) and g(x).
(15) Suppose F 0 (x) = f (x), F (0) = 0, F (1) = 1 = F (e), u(0) = 0, u(2) = 1, and
compute
Z the following integrals
2
f (u)u0 dx
(a)
Z0 π/2
(b)
f (sin x) cos xdx
0
2
Z
1
(c)
f (ex )ex dx
Z0 e
f (ln x)
dx
x
1
(16) Consider the matrix
(d)
(17)
(18)
(19)
(20)
(21)
1 2 3
A=
.
4 5 6
(a) What are the dimensions of A?
(b) What is a2,1 ?
(c) Compute 5A.
Suppose a matrix has dimensions 3 × 5.
(a) How many rows does this matrix have?
(b) How many columns does this matrix have?
(c) How many elements does this matrix have?
Let
0 1 3
B=
.
1 0 6
What is A + B?
Does the product AB exist?
Let


0 1
C =  1 0 .
2 3
What is AC?
Solve the system by elimination

 x + 2y + 4z = 5
y−z =6

x+z =1
13 22 8
, ,− )
5 5
5
(22) Solve the system by elimination

 2x + 2z = 5
y−z =6

x+z =1
Solution (
This system has no solutions.
(23) Solve the system by elimination

 2x + 2z = 2
y−z =6
 x+z =1
This system has infinitely many solutions: (1 − r, 6 + r, r), where r is any real
number.
3