CALCULUS 1000 B — WINTER 2023
Assignment 3 (template)
Due Date: Monday, April 10th at 11:55 p.m.
Total: 50 marks
INSTRUCTIONS
Each question should be submitted on Gradescope (www.gradescope.ca) with your handwritten
work clearly shown, and it is expected that homework is done individually. Your work should refer
to the material learned in this course and the material presented in your previous math courses.
Remember to justify your calculations and conclusions. A poorly justified solution will not receive
many marks. A solution with just a final answer and no work shown will receive 0. Be sure to
clearly state your reasoning (“since the function satisfies the conditions of...”, “...which we evaluate
using the identity...”, etc.).
Name
Jeremy Ro
Student ID (print)
251319988
Q1. (3 marks) Find the function f (x) if f 00 (x) = 2 + 12x  12x2 and f (1) = 5, f 0 (1) = 2.
fl
1215
x
1211 2
E ZI
y
t
a an
2
2
21 17
12
6
44
2
f
t
c
5
s
age
IIIC
x
2
II
2
one
12
41
15
10
f
x
2
1221 1231 10
fac 221 1211 14
1
1071
15
lol
t
C
CALCULUS 1000B, Winter 2023
Assignment 3
Q2. (6 marks)
(a) Evaluate the definite integral,
0
suitable Riemann sum.
I
315 4 1 dx
xi
ftpgf xi lsx
self
se
at
a i
Ot
f
Z
1
Iti
I
1 41
IE
ant
fI I 3,9Mt
t.fi
f
fine
t
i
ing
n
an
2
4E I
864 1 3

3x2  4x + 1 dx by calculating it as a limit of a
f
I
intodefinition
267 1
2
(b) In this question, write the given sum in Sigma notation as the Riemann sum for a
Z b
function and express the limit as a definite integral
f (x)dx:
a

 
 



2
2
4
6
2n
lim
+1 +
+1 +
+ 1 + ... +
+1
n!1 n
n
n
n
n
Then evaluate the integral.
s 864dm
a
E
sKf ail
fly general formula
Die
Jey
Baby
212 2
6 2
f
Guild
x
1
Ntl
but
r
4
2
1
0
4
CALCULUS 1000B, Winter 2023
Assignment 3
Q3. (6 marks) Solve the following optimization problems. Be sure to show your work and to
justify any domains of definition for your functions
(a) A rover on Mars has just collected a very interesting soil sample. It will launch a two
stage rocket to send the sample to an orbiting satellite for further analysis. The distance
above the surface of Mars that the rocket carrying the sample is, after t hours, is given by
f (t) = t3  9t2 + 24t kilometers. The first stage of the rocket has enough fuel to burn for
exactly 6 hours. The rocket’s second stage should be fired during this 6 hour window when
the rocket’s height is maximized. When should the second stage be fired?
f
2
P
f
814 83 97 241
2
9212 2462
2
2,20
20
2
841 3 181 24
0 3
2
f
18 24
0 3152 68 8
34 2 t Y
2 2 2 4
Y
Max
43 91411 24 4
4,11
16
min
The rochetshouldbefired when 4 2
aka Thus
(b) A 1 m length of wire is to be cut into two pieces. The first piece will be bent to form
a circle, while the second piece will be bent to form an equilateral triangle. Maximize the
total area enclosed by these two shapes.
Max area cut should be made
pg
A Aq AD
X
HE
flat
1 74
A
ta
O
Et
esta CE F
T
ink il
TB 9K TAX
Ex
3
gIFÉ
CALCULUS 1000B, Winter 2023
Assignment 3
Q4. (4 marks) A particle moving in a straight line with a velocity (in m/sec) given by v (t) =
t2  2t  3. Determine
(a) the displacement of the particle for 2  t  4 seconds;
(b) the distance travelled by the particle for 2  t  4 seconds.
d
a integral of it
d
j 27 3 at
f
It'dt
1
It
2
El
It
I
I
at
2 3 It
2
III
that
on
If
3
314 3637
g
seconds
5 3
3 3
4
3,1 y y
4m
by expressing this integral in terms of the function F.
g
Sitcom Bida
3 Skokie 3 idk
I f d deal
x
1 I ZS
Displacement
Q5. (4 marks) Suppose we know that F 0 (x) = x cot (x) . Simplify
Z
 
x5 cot x3 dx
Iz F
t 27 3dt
t
S 3
142 3
21214
f
3at Itt
Sitt f
2
It't
34
531 12 6
displacement
1
d s
Fbi
a Fadl
sicotte
x'colfi
x'corn 310dx
C
4
did
Distance
4m
M
CALCULUS 1000B, Winter 2023
Assignment 3
Q6. (9 marks)
d
(a) Find the derivative,
dx
Z
x
0


ln tan(t)  arctan(t) dt.
x'f nG4
Iln tanks arctank
f'ay ln
tank tan 64
d
(b) Find the derivative,
dx
41,48Hdt
flag
Z
ex
x2
sin
p

3
t dt.
421648122cm
e
singer
hi fi f hi
x
since
2x
(c) Determine the intervals over which the function F (x) =
Z
x
3
et dt is concave up and
0
concave down.
I
fetate
j
t
t
g
81164 371
U
V
314 0
0
2 301
2
0
5
It is coniamuponte
interval a x
CALCULUS 1000B, Winter 2023
Assignment 3
Q7. (9 marks) Each of the following definite integrals can be computed through substitution.
Make sure to clearly state the function you are substituting, and clearly indicate how the
limits of integration
of the definite integral change when you perform the substitution.

Z e sin  ln(x)
3
(a)
dx
x
1
Ian
sine
Stsinfidu
(b)
Z
ln(7)
p
ex 2 + ex dx
ln(2)
IF
Eiji
(c)
1
3
x2
o
E
f
cost
cost
ios o
Kent vatemaleg
71 Inf V yeen21 4
I
du
3643 3 347
Z
E
1
dx
+ 4x + 5
EY
18
33
4
II
It
Fit
Into
tan
u
Janki sanity
Iz
6
CALCULUS 1000B, Winter 2023
Assignment 3
Q8. (9 marks)
(a) Evaluate the sum
11
X

i=1
IFi
3
É
5
i
4411144
11
1

i2 + 3i  5 .
3611451
Y
3
2 237 3166
11 5
55
55
649

101 
X
1
1
(b) Evaluate the sum

.
i+1 i+3
i=1
z
I
4
t
t
t
t
t
Gotz
h
(c) In terms of m and n (with 1  m < n), find a formula for:
m
I
É
il
m
M 3
If
nth
at
it
If
tf
I
inning
7
n
X
i2 .
i=m
natty
5
g
2
it
Ky
EDY
is
D
nfntnrytil
mcm
n.tl