MCV4U1 – Spring 2022 (PM)
/ 13 (K)
Bur Oak Secondary School
Mathematics Department
/ -- (T)
/ 10 (A)
/ 7 (C)
Test – Limits & Rates of Change (Part 1)
Name: ___________________________________________________________________________________
Note: Solutions must be presented using ideas and concepts discussed in the current unit of study for full credit.
[3]
Mathematical form used throughout this assessment.
Form Success Criteria
¨ All steps are shown in an organized and easy to follow manner.
¨ Appropriate symbols and notation are used consistently and correctly. (brackets, fraction lines, equal signs etc...)
¨ Final answers are written clearly with appropriate units and as sentences for word problems.
Part A – Knowledge
1) Consider the graph of the function ! = #(%) below. Given the following questions, state your answer in the space
provided.
[6 x 0.5 marks]
a)
b)
c)
lim #(%)
!→#!
lim #(%)
!→$%"
lim #(%)
!→&"
d) lim #(%)
!→&
e) #(3)
f)
#′(−4)
8
U
ar
-
Iv
ooo
V
PNE
-
I
DNE
.
✓
V
Figure 1 - y = f(x)
Page 1 of 4
MCV4U1 – Spring 2022 (PM)
2) Calculate the value of each of the following limits (if they exist):
a)
lim
! ! #%!#&'
[3]
$!()
!→#$
b) lim
iii. i:¥¥
x'Is
'
=
√!#'#$
!→'*
'
¥
'
'
"
#&
#
.IT/t3fxTt3
[3]
'
II
.
'
x'Ino
-10/9
-
Is
'
=
&
-2
"
Eno
516
'(
3) Let #(%) = ! # . Determine '! by first principles.
84N
[4]
Itt Fa
Io
-
T
'
.
i :O 's
'
f ' 4) =-D
xs
"
n'Intl "¥fht
''
'
-
)
't:O #3*-17×14,1-32
=
iii. ÷
)
limo 's,
-
-
lim
=
-
line
-
-
-
.
=
-
=
x
-
3h
347¥77
-6€
x'(x)
-
6
Text
h -70
-5¥
'
Page 2 of 4
MCV4U1 – Spring 2022 (PM)
Part B – Application
4) A rock is thrown from a cliff. The height of the rock (in m) . seconds after it is thrown is given by
/(.) = 20. − 4.9. ) .
a) Calculate /′(.) using first principles.
H' ( t)=lim20Lxth )
h
=
-
[4]
4.9 ( xth )
'
-
( 20T
-
4.9T )
#
so
-
20XI-20h-4.9lxzt2xh.tt#-2Oft4.qth-so
lim
h
=
2otttaoh-4.at/-a.sth-#h-2/ott4.9tT
no
h
slim
h -70
20h
-
9. 8th
-
4.942
I
KL-20-a.si#--49h)=lihhfol2O-9.8t-U.9h
=
thing
,
)
20
=
i.
'*
b) Calculate '+ 4
+,)
9.
-
h' LH
20 -9.8T
-
and describe its significance.
20-9.8127=215
This
set
represents
[2]
m/s
the
Inst
.
speed
of
the
ball
at
2. Sees
.
Page 3 of 4
MCV4U1 – Spring 2022 (PM)
−7% + 48) , % > 1
5) Let #(%) = 5
where 8 ∈ ℝ. Determine the exact value(s) of 8 such that #(%) is continuous at #(1).
2% + 8, % ≤ 1
[4]
¥7
( 2x
ta
)
C- txt Haz)
; Y
.
,
2
t a
U
•
t
-
4
=
a=
•
⑨
=
a
4aZ
t
2
-
a
-
9
ltg
a
,
=
I
8
Part C – Communication
6) Explain how the calculus method of calculating the instantaneous rate of change differs from the advanced functions
method. Use an example to assist your explanation.
[4]
In
Advanced functions
fCataoDparticular point
and
would
slope
of
value
.
that
we
used
we
r
a
r
r
4
wanted for example
,
value
that
and
814.017-61410
calculus
where
which
approaching
-
In
in
r
Complete explanation provided with
appropriate example
Partial explanation provided with appropriate
example
Partial explanation provided with no example
Appropriate use of mathematics terminology
the
find the difference between
values
E.g
,
did
we
the
O
the
l
we are
value
at
doing him
of
h
is
f¥
approaching
in: ::c: :
,
zero
.
.
Page 4 of 4