Mathematics 2122-002
Calculus for Life Sciences II
Fall 2013
Final Guide
Instructor: Dr. Alexandra Shlapentokh
(1) If 2 sin x is an antiderivative of f (x), then what is f (x)?
(2) Suppose an antiderivative of f (x) is sin(ex ). Describe all the other antiderivatives
of y = f (x).
(3) If an antiderivative of y = f (x) is tan(3x + 1) and an antiderivative of y = g(x) is
2
ex , then what’s an antiderivative of
(a) 2f (x)?
(b) f (x) + g(x)?
(4) What are the antiderivatives of the following functions?
(a) y = sin x
(b) y = ex
(c) y = e−2x
(d) y = cos x
(e) y = sec2 x
(f) y = sec2 3x
(g) y = csc2 x
(h) y = csc2 (−2x)
(i) y = x2 + 5x + 6
1 + 3x + 3x2
(j) y =
5x
(k) y = x1
(l) y = cos 2x
(m) y = sin πx
(5) 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?
(6) 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?
(7) 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?
(8) Suppose y = f (x) is a function continuous everywhere and is such that f (x) < 0
for x ∈ [1, 2), f (2) = 0 and f (x) > 0 for x ∈ (2, 3]. Express the area bounded by the
graph of y = f (x), x-axis, and vertical lines x = 1 and x = 3 in terms of a definite
integral of f (x).
(9) Compute
Z 4 the following definite integrals.
1 + 3x + 3
dx
(a)
5x
2
1
Z
6
(b)
Z5
(c)
9
√
2xdx
1
dx
x
Z8 π/2
(d)
sin xdx
Z0
(e)
π/2
cos xdx
0
(10) 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.
(d) Find the area bounded by y = −x2 + 9 and the x-axis between x = −3 and
x = 3.
(e) Find the area bounded by y = sin x and the x-axis between x = 0 and x = π.
(f) Find the area bounded by y = sin x and the x-axis between x = 0 and x = 2π.
(g) Find the area bounded by y = cos x and the x-axis between x = 0 and x = π.
(h) Find the area bounded by y = cos x and the x-axis between x = 0 and x = 2π.
(11) 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]
(12) Suppose f 0 (x) = x2 − x1 and f (1) = 0. Find a formula for f (x). Solution: f (x) =
x3 /3 − ln |x| + C, 0 = 13 − ln 1 + C, C = − 13 .
(13) Suppose a particle moves along a straight line with a(t) = t3 . Find the position
function if v(0) = 3 and s(0) = 0. Solution: v(t) = t4 /4 + C, 3 = v(0) = C,
v(t) = t4 /4 + 3, s(t) = t5 /20 + 3t + C, 0 = s(0) = C, s(t) = t5 /20 + 3t + C.
(14) What are the antiderivatives of the following functions?
(a) y = sin2 x cos x
(b) y = cos2 sin x
3
(c) y = x2 ex
(d) y = sin(3x + 1)
ln2 (x)
(e) y =
x
(f) y = tan x
(g) y = cot x
(15) Compute
Z 6 the following definite integrals.
√
(a)
2x + 1dx
5
2
Z
100
x+1
dx
x2 + 2x
10
Z 1
2
ex
(c)
dx
1 ln x + 1
Find the following areas.
(a) The area bounded
by the curves y = −x2 + 9 and y = x2 − 9.
R3
Solution: −3 (−x2 +9)−(x2 −9)dx = 18x− 23 x3 |3−3 = 54− 32 ·27+54− 23 ·27 = 72.
(b) The area bounded by sin x, cos x between x = 0 and x = π/4. Solution:
√
√
√
R π/4
π/4
cos
x
−
sin
xdx
=
sin
x
+
cos
x|
=
1/
2
+
1/
2
−
1
=
2−1
0
0
(c) The area bounded by sin x, cos x between x = 0 and x = π/2.
R1
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 ofRf (x) and g(x) between
x = −1 and Rx = 1.
R −1
−1
−1
Solution −1 (f (x) − g(x))dx = −1 f (x)dx − −1 g(x)dx = 7 − 3 = 4
R1
Let f (x), g(x) be two continuous functions on [−1, 1] with −1 f (x)dx = 7. Suppose
the area
R 1 bounded by the graphs of f (x) and g(x) between x = −1 and x = 1 is 5.
Find −1 g(x)dx.
Solution: there is not enough information to answer the question. One need to
know the relative positions of f (x) and g(x).
Let f (x, y, z) = x2 y−3z+5. Compute f (3, 2, 2). Solution: f (3, 2, 2) = 9·2−3·2+5 =
17.
p
Find the domain of f (x, y) = x2 − yx. Solution: {(x, y) ∈ R2 : x2 − yx ≥ 0}.
x2
. {(x, y) ∈ R2 : yx 6= 1}
Find the domain of f (x, y) = yx−1
(b)
(16)
(17)
(18)
(19)
(20)
(21)
2
x
(22) Find the domain of f (x, y) = √yx−1
.Solution: {(x, y) ∈ R2 : yx − 1 > 0}
(23) Compute ∂f /∂x, ∂f /∂y, ∂ 2 f /∂x2 , ∂ 2 f /∂y 2 , ∂ 2 f /∂x∂y for the following functions:
(a) f (x, y) = x ln y + x sin y − exy
(b) f (x, y) = (x + y)20 (cos(xy) − sin(x − y)).
Partial solution: ∂f /∂x = 20(x+y)19 (cos(xy)−sin(x−y))+(x+y)20 (−y sin(xy)−
cos(x − y)), ∂f /∂y = 20(x + y)19 (cos(xy) − sin(x − y)) + (x + y)20 (−x sin(xy) +
cos(x−y)), ∂ 2 f /∂x2 = 20·19(x+y)18 (cos(xy)−sin(x−y))+20(x+y)19 (−y sin(xy)−
cos(x − y)) + 20(x + y)19 (−y sin(xy) − cos(x − y)) + (x + y)20 (−y 2 cos(xy) +
sin(x − y)).
(c) f (x, y, z) = xyz tan(x + y + z)
(d) f (x, y, z) = xyz − x − y − z
(24) What is a general solution to the equation y 0 = f (x), where f (x) is a continuous
function?
R
y = f (x)dx
(25) Find a particular solution to the equation y 00 = x3 , if y 0 (1) = 0 and y(2) = 0.
4
We integrate y 00 to conclude that y 0 = x4 + C. Substituting 1 for x and 0 for y we
4
obtain 0 = 14 + C and conclude C = − 14 . Thus y 0 = x4 − 14 . Integrating y 0 we
5
32
obtain y = x20 − 14 x + C. Substituting 2 for x and 0 for y we obtain 0 = 20
− 21 + C.
5
11
11
Therefore, C = − 10
and y = x20 − 14 x − 10
.
3
(26) Solve y 0 − 5x2 y = x2 , y(1) = 2.
(27) Solve xy 0 − 4y = x, y(2) = 3
R
Rewrite the equation y 0 − x4 y = 1. Note that p(x) = − x4 , F (x) = − x4 dx = ln x−4 +C,
−4
G(x) = eln x = x−4 . Next we have (yx−4 )0 = x−4 , yx−4 = − 31 x−3 + C, y =
− 13 x + Cx4 . Substitute y = 3, x = 2 to get 3 = − 23 + 16C, C = 11
. Final answer is
48
11 4
x.
y = −3x + 48
(28) Suppose a tank contains 200 gal of water and 7 pounds of a certain chemical. A
solution of the same chemical containing 3 pounds per gallon is being poured into
the tank at the rate of 5 gallons 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. Determine the
formula for A(t) and limt→∞ A(t). (Assume the distribution of the chemical in the
tank is uniform at any moment of time.)
(29) What is the general solution to the equation y 0 = ky?
(30) Solve y 0 = 0.1y, y(1) = 2.
(31) Solve y 0 = 3xy, y(0) = 5.
3
(32) Solve y 0 = 2et t2 y.
et
(33) Solve y 0 = 2
y +y
(34) Higher Order Homogeneous Differential Equations: Examples 1, Example 5, Example 8, pages 598–602
(35) Solve y 00 − 5y 0 − 6y = 0, y 0 (0) = 1, y(0) = 1
(36) Solve y” − 2y 0 + 1 = 0, y 0 (0) = 1, y(0) = 1
(37) Solve y 00 + 16y = 0.
4