MAT186H1F  Calculus I
Fall 2023
During the first 25 minutes of your (first) tutorial next week, you will complete a twoproblem tutorial
worksheet (possibly with parts) drawn from these problems. You may discuss the problems in groups but
the writeup and submission of your worksheet must be completed individually.
During the remaining 25 minutes of your tutorial, you will discuss in groups any of the problems included
on this sheet. They are meant for your own practice. You do not need to complete them all before your
tutorial but please bring some of your draft work and a willingness to share your ideas or listen intently.
You may also use the remaining time of your tutorial to discuss Assignment 1.
Throughout your tutorial, TAs will circulate throughout the rooms as you work in small groups providing
hints and feedback on your preliminary ideas.
I. Building Blocks
1.1 Vocabulary
1. Recall the definitions for the following terms to determine if the statements below are TRUE or FALSE.
Provide a sentence or two that either correct the statement (if
false
) or justify the statement (if
true
).
(a) sin
θ
is the
x
coordinate of the point on the unit circle that is reached after traveling the distance
and direction indicated by the angle
θ
, from the point (1
,
0).
(b) sin

1
(
x
) yields radian measures between 0 and
π
2
(inclusive) or
3
π
2
and 2
π
(inclusive) for which
sine equals
x
.
(d) Given a transformed sine function,
g
(
x
) =
c
·
sin[
a
(
x
+
b
)]+
d
, the amplitude,

c

, can be determined
by calculating half of the distance between the maximum and minimum values of the function.
(i.e.

c

=
max

min
2
).
(e) The function
f
(
x
) = sin
x
+ 1 is neither even nor odd.
1.2 Fundamentals and Basic Procedural Exercises
2. Determine if the following statements are TRUE or FALSE with a short justification.
(a) Given an invertible function
f
and its inverse
f

1
,
f

1
(
f
(
x
)) =
x
for all
x
∈
R
.
(b)
√
sin
2
x
= sin
x
for all
x
∈
R
.
(c) The function
f
(
x
) = 2 cos(3
x

6) + 1 has an amplitude of 2, vertical shift of 1, a period of 2
π
,
and a horizontal shift 6 units to the right.
(d)
e
ln
x
+1
=
x
+ 1 for all
x
∈
R
.
(e) The domain of ln(
x
2
) is (
∞
,
∞
).
(f) If
g
(
x
) = 5
x
and
h
(
x
) = 8
2
x
, then
f
(
x
) =
g
(
x
)
h
(
x
) is an exponential function.
3. Show that the only values of
x
that satisfy
x
log
10
x
= 100
x
are
x
= 100 or
x
=
1
10
.
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of
3