Math 1210: Worksheet #2
Winter 2017
Name:
A#:
Section:
1. (2) In the coordinate system below sketch the graphs of functions y = 1 + sin(x + π2 ) and
y = −1 + sin( x2 ). Make sure that you clearly label each curve.
y
2
1
−π
π
2
−π
2
π
3π
2
2π
5π
2
x
-1
-2
2. (3) In the coordinate system below sketch the graphs of the functions y = 1+ex , y = ln(x),
and y = tan−1 (x). Make sure that you clearly label each curve and also the y-axis. (For
a more realistic special effect, use the fact that tan−1 (3.7) ≈ ln(3.7) ≈ 1.3)
y
-3
-2
-1
1
2
3
4
x
3. (3) Find the equation of the secant line of the curve y = x2 + x + 1 over the interval [1, 2].
v2.1
4. (12) Perform the required task (if any) and simplify. For questions (k)-(n) assume that
f (x) = x2 + 2x and g(x) = x + 1.
(a) Write as a sum of summands of the form xa .
(b) Put on common denominator.
(c)
1
x+1
−
1
x
√
x3 + 3 x
2
x
=
=
√
x2 + 4x + 4 =
(d) Expand. (x2 + x + 1)(x2 − 2x + 3) =
(e) Assume x 6= −6.
x2 +5x−6
x+6
=
(f) Rationalize. Assume x 6= 1. √
(g)
(ex )x−1 ex−1
e3
x2
x−1
√ =
+1− 2
=
(h) ln(2ex /3) + ln(3ex ) − ln(2) =
(i) f (f (x)) =
(j) f (g(x)) =
(k) g(f (x)) =
(l) Assume that the domain of f is [−1, ∞). f −1 (x) =
Bonus. (4)
(a) sin−1 (sin( 10π
)) =
3
(b) Assume x 6= ±1. tan(sin−1 (x)) =
v2.1
Math 1210: Worksheet #2
Winter 2017
Name:
A#:
Section:
1. (2) In the coordinate system below sketch the graphs of functions y = −1 + cos(x + π2 )
and y = 1 + cos( x2 ). Make sure that you clearly label each curve.
y
2
1
−π
π
2
−π
2
π
3π
2
2π
5π
2
x
-1
-2
2. (3) In the coordinate system below sketch the graphs of the functions y = 1+ex , y = ln(x),
and y = tan−1 (x). Make sure that you clearly label each curve and also the y-axis. (For
a more realistic special effect, use the fact that tan−1 (3.7) ≈ ln(3.7) ≈ 1.3)
y
-3
-2
-1
1
2
3
4
x
3. (3) Find the equation of the secant line of the curve y = x2 − x + 1 over the interval
[−1, 2].
v2.2
4. (12) Perform the required task (if any) and simplify. For questions (i)-(l) assume that
f (x) = x2 − 2x and g(x) = x + 1.
(a) Write as a sum of summands of the form xa .
(b) Put on common denominator.
(c)
1
x−1
−
1
x
√
x2 + 5 x
3
x
=
=
√
x2 − 4x + 4 =
(d) Expand. (x2 − x − 1)(x2 + 2x + 3) =
(e) Assume x 6= −2.
x2 +7x+10
x+2
=
x+3
(f) Rationalize. Assume x 6= −3. √ 2
=
x +x+3−3
(g)
(ex )x−2 ex−1
e2−3x
=
2
(h) ln(3ex /2) + ln(2ex ) − ln(3ex ) =
(i) f (f (x)) =
(j) f (g(x)) =
(k) g(f (x)) =
(l) Assume that the domain of f is [1, ∞). f −1 (x) =
Bonus (4)
(a) cos−1 (cos( 10π
)) =
3
(b) Assume x 6= ±1. sec(sin−1 (x)) =
v2.2
Math 1210: Worksheet #3
Winter 2017
Name:
A#:
Section:
1. (5) Let f be a function whose graph of y = f (x) is given below.
y
6
tangent
4
2
-10
-5
5
10
15
20
25
x
-2
(a) lim f (x) =
x→15
(b) lim+ f (x) =
x→10
(c) List all values of a in the interval (−15, 30) where the limit x→a
lim f (x) does not exists:
(d) The average rate of change of f (x) over the interval [0, 10] is
(e) The instantaneous rate of change of f (x) when x = 15 is
√
2. (5) Let f (x) = 3x and let a > 0. Find the instantaneous rate of change of f (x) when
x = a, as a function g(a) of a (you are not allowed to use any theory of derivatives).
v3.1
3. (10) Compute the limit or show that it does not exists.
x2 − x − 6
(a) lim 3
x→3 x − 9x
(b) lim+
√
x→1
(c) lim
x→2
√
√
x + 1 − x2 + 1
x2 + 3x − 4
x2 + x + 3 −
|x − 3|
√
x+2
x2 − 4
x→−2 |x + 2|
(d) lim
x2 − 2x + 1
x→1
|x − 1|
(e) lim
v3.1
Math 1210: Worksheet #3
Winter 2017
Name:
A#:
Section:
1. (5) Let f be a function whose graph of y = f (x) is given below.
y
9
tangent
6
3
-4
-2
2
4
6
8
10
x
-3
(a) lim f (x) =
x→8
(b) lim+ f (x) =
x→4
(c) List all values of a in the interval (−6, 12) where the limit x→a
lim f (x) does not exists:
(d) The average rate of change of f (x) over the interval [0, 2] is
(e) The instantaneous rate of change of f (x) when x = 6 is
√
2. (5) Let f (x) = x + 1 and let a > 0. Find the instantaneous rate of change of f (x) when
x = a, as a function g(a) of a (you are not allowed to use any theory of derivatives).
v3.2
3. (10) Compute the limit or show that it does not exists.
x3 − 4x
(a) lim 2
x→2 x + 4x − 12
(b) lim− √
x→1
(c) lim
x→1
√
x2 − 5x + 4
√
1 + x − x2 + 1
x2 + x + 2 −
|x − 4|
√
x+8
x2 − 1
x→1 |1 − x|
(d) lim
x2 + 4x + 4
x→−2
|x + 2|
(e) lim
v3.2
Math 1210: Worksheet #4
Winter 2017
Name:
A#:
Section:
1. (6) Draw a graph of y = f (x) for a function f satisfying all of the following:
3
• x→∞
lim f (x) = , lim f (x) = −1
2 x→−∞
• lim f (x) = −∞, lim− f (x) = ∞, lim + f (x) = −∞, lim − f (x) = ∞
x→3
x→1
x→−2
x→−2
1
• lim+ f (x) = f (1) = , lim+ f (x) = f (2) = −1, lim− f (x) = 1.
x→1
x→2
2 x→2
y
2
1
-3
-2
-1
1
2
3
4
x
-1
2. (2) List vertical asymptotes of y =
x−1
+ ln |x2 − 4|.
x2 − 1
√
x2 + x + 1
3. (4) Find horizontal asymptotes of y =
x+1
v4.1
4. (8) Compute the limit. Any 4 questions are for a full mark. The remaining question is
for 2 bonus marks.
2 sin t
sin x
+ lim
t→−∞
x→0 2x
t
(a) lim
(b) x→∞
lim
√
(c) lim
x→−∞
e2x − ex + 4 − ex
ex + x3 − 2x2
x3 − x2 + 2x − 1
(d) x→∞
lim
2x − 3x2
−1
(e) lim+ tan
x→1
x+2
x−1
v4.1
Math 1210: Worksheet #4
Winter 2017
Name:
A#:
Section:
1. (6) Draw a graph of y = f (x) for a function f satisfying all of the following:
3
5
• x→∞
lim f (x) = , lim f (x) = −
2 x→−∞
2
• lim f (x) = −∞, lim− f (x) = ∞, lim + f (x) = −∞, lim − f (x) = ∞
x→3
x→2
x→−2
x→−2
1
• lim+ f (x) = f (2) = 1, lim + f (x) = f (−1) = − , lim − f (x) = 2.
x→2
x→−1
2 x→−1
y
2
1
-3
-2
-1
1
2
3
4
x
-1
2. (2) List vertical asymptotes of y =
x−2
+ ln |x2 − x|.
x2 − 4
2x − 1
3. (4) Find horizontal asymptotes of y = √ 2
x −x+2
v4.2
4. (8) Compute the limit. Any 4 questions are for a full mark. The remaining question is
for 2 bonus marks.
sin t
2 sin x
+ lim
t→−∞ 2t
x→0
x
(a) lim
(b) x→∞
lim ex −
(c) lim
x→−∞
√
e2x − 3ex + 8
e−x − x3 − 2x2
x3 − x2 + 2x − 1
(d) x→∞
lim
2x − 3x4
−1
(e) lim+ tan
x→1
x−2
x−1
v4.2
Math 1210: Worksheet #5
Winter 2017
Name:
A#:
Section:
1. (6) Let f be a function whose graph of y = f (x) is given below.
y
y = f (x)
3
2
tangent
1
-1
1
2
3
4
5
6
x
Fill in the following.
(a) List all x in the interval (−2, 7) where f is not continuous:
(b) List all x in the interval (−2, 7) where f is not left-continuous:
(c) List all x in the interval (−2, 7) where f is not differentiable:
(d) lim+ f (et + 5) =
t→0
(e) f 0 (4) =
(f) If g(x) =
f (2x)
, then g 0 (2) =
2
x +1
2. (2) Find a function f and a number a such that
e3+h + tan−1 (2(3 + h) + 1) − e3 − tan−1 (7)
h→0
h
f 0 (a) = lim
a=
f (x) =
3. (3) Find the equation of the tangent line to y = e1−3x at x = 1.
v5.1
4. (3) Find values of a and b such that the function
 sin x

 x
, for x < 0
, for x = 0
f (x) = 2a
 x

e + b , for x > 0
is continuous everywhere.
5. (6) Compute the derivative. Do not simplify.
(a)
d 7
x 3 sin(x)ex
dx
d
(b)
dt
(2t + 1)2 cos(5t)
t−2 + 3 tan(t)
!
v5.1
Math 1210: Worksheet #5
Winter 2017
Name:
A#:
Section:
1. (6) Let f be a function whose graph of y = f (x) is given below.
y
y = f (x)
3
2
tangent
1
-1
1
2
3
4
5
6
x
Fill in the following.
(a) List all x in the interval (−2, 7) where f is not continuous:
(b) List all x in the interval (−2, 7) where f is not right-continuous:
(c) List all x in the interval (−2, 7) where f is not differentiable:
(d) lim+ f (6 − 2t) =
t→0
(e) f 0 (4) =
(f) If g(x) =
f (3x − 2)
, then g 0 (2) =
x2 − 1
2. (2) Find a function f and a number a such that
e2+h + cos−1 (3(2 + h) + 2) − e2 − cos−1 (8)
h→0
h
f 0 (a) = lim
a=
f (x) =
3. (3) Find the equation of the tangent line to y = cos(3x) at x = π4 .
v5.2
4. (3) Find values of a and b such that the function
 sin x

 x
is continuous everywhere.
, for x > 0
, for x = 0
f (x) = 3a


ln(2 − x) + b , for x < 0
5. (6) Compute the derivative. Do not simplify.
(a)
d 5
x 3 cos(x)ex
dx
d
(b)
du
(3u + 1)3 cos(4u)
√
u + 3 sec(u + 1)
!
v5.2
Math 1210: Worksheet #6
Name:
Winter 2017
A#:
Section:
1. (12) Compute the derivative. Do not simplify.
(a)
d (sin(u2 ))2 e
du
(b)
d ln(ex + 1) sin−1 (2x + 1)
dx
(c)
d x
x
(e + 1)e
dx
(d)
d ln(t2 + 1)
dt t2 + 1
v6.1
2. (4) Find the equation of the tangent line to the curve given by x2 y 2 = x2 − x + 2 at the
point (2, 1).
3. (4) Let x, y be functions of t related by
ex
Compute
dy
dt
2 +y
= x2 + 2 + sin(y 2 ).
in terms of x, y, dx
.
dt
(
x2 sin(1/x) , for x 6= 0
. Compute f 0 (0) and show that
0
, for x = 0
lim f 0 (x) does not exist (which shows that f is everywhere differentiable, but f 0 is not
x→0
continuous at 0).
Bonus Question.[6] Let f (x) =
v6.1
Math 1210: Worksheet #6
Name:
Winter 2017
A#:
Section:
1. (12) Compute the derivative. Do not simplify.
(a)
d (cos(u2 ))2 2
du
(b)
d ln(ex + 1) tan−1 (x2 + 1)
dx
(c)
d x
(e + 1)tan(x)
dx
(d)
d ln(t2 + 1)
dt t2 + 1
v6.2
2. (4) Find the equation of the tangent line to the curve given by xe3x−y = x2 − 2y + 5 at
the point (1, 3).
3. (4) Let x, y be functions of t related by
tan−1 (x2 + y) = x2 y
Compute
dy
dt
in terms of x, y, dx
.
dt
(
x2 sin(1/x) , for x 6= 0
. Compute f 0 (0) and show that
0
, for x = 0
lim f 0 (x) does not exist (which shows that f is everywhere differentiable, but f 0 is not
x→0
continuous at 0).
Bonus Question.[6] Let f (x) =
v6.2
Math 1210: Worksheet #7
Name:
Winter 2017
A#:
Section:
1. (6) Consider a right-angled triangle with catheti (adjacent sides) x and y and hypothenuse
(opposite side) z. When x = 5 and y = 12 we have that x is growing at the rate of 2cm/s,
y is decreasing at the rate of 1cm/s (the right angle π2 between x and y is fixed throughout
the process). Answer one of the following questions (circle your choice). You can also
answer both questions for 4 bonus marks, however no part marks will be awarded in
the bonus part.
(a) What is the rate of change of the angle θ between x and z?
(b) What is the rate of change of z?
2. (3) A snow ball of radius r is thrown into warm water. Its volume V is decreasing at a
= −kS, where k is a positive constant. Find
rate proportional to its surface S, i.e., dV
dt
the rate of change of radius r (in terms of k).
v7.1
3. (3) Let f be a function whose derivative is
f 0 (x) =
(x + 2)4 ln(x2 )ex
√
.
3
x+9
List critical values and the intervals where f is increasing.
4. (8) Find the global maximum and the global minimum of f (x) = (x2 − 2x)ex on the
interval [0, 3]. For 5 bonus marks find the global maximum and minimum of f on
(−∞, 0] or explain why they do not exist.
v7.1
Math 1210: Worksheet #7
Name:
Winter 2017
A#:
Section:
1. (6) Consider a right-angled triangle with catheti (adjacent sides) x and y and hypothenuse
(opposite side) z. When x = 5 and z = 13 we have that x is growing at the rate of 2cm/s
and z is decreasing at the rate of 1cm/s (the right angle π2 between x and y is fixed
throughout the process). Answer one of the following questions (circle your choice). You
can also answer both questions for 4 bonus marks, however no part marks will be
awarded in the bonus part.
(a) What is the rate of change of the angle θ between x and z?
(b) What is the rate of change of y?
2. (3) A snow ball of radius r is thrown into warm water. Its volume V is decreasing at a
= −kS, where k is a positive constant. Find
rate proportional to its surface S, i.e., dV
dt
the rate of change of radius r (in terms of k).
v7.2
3. (3) Let f be a function whose derivative is
f 0 (x) =
(x + 2)4 ln(x2 )ex
√
.
3
x+9
List critical values and the intervals where f is increasing.
4. (8) Find the global maximum and the global minimum of f (x) = (x2 − 2x)ex on the
interval [0, 3]. For 5 bonus marks find the global maximum and minimum of f on
(−∞, 0] or explain why they do not exist.
v7.2
Math 1210: Worksheet #8
Name:
Winter 2017
A#:
Section:
1. (9) Find the point(s) on the parabola y = x2 that are closest to the point (0, 1). Hint:
minimize the square of the distance.
v8.1
2. (11) Alice an Bob are on opposite side of a slow moving river that is 120 ft. wide. Bob is
L ft. downstream from Alice. He has hurt his leg and Alice wants to help. Alice swims
at 3ft./s and walks at 5ft./s. Plot a route for Alice that will get her to Bob in shortest
time possible (with no loss of generality assume that Alice will swim before she does any
walking).
(a) L = 160 ft.
(b) L = 50 ft.
v8.1
Math 1210: Worksheet #9
Name:
Winter 2017
A#:
Section:
1. (6) Compute the limit and explain why L’Hôpital’s rule does not apply.
x + cos x
lim
x→∞ x + ln x
L’Hôpitals rule does not apply because:
2. (6) Compute the following limits.
ex − 1 − x
x→0 x sin(x)
(a) lim
(b) lim+
x→0
√
x ln(x)
v9.1
3. (4) Find the linearisation L(x) of f (x) =
√ x−2
2xe
centred at 2.
4. (4) Use linearisation to find a reasonable approximation of
√
14.
Bonus. (5) Let a, L be a real numbers and let f be a continuous function such that lim f 0 (x) = L.
x→a
Prove that f is differentiable at a and that f 0 (a) = L.
v9.1
Math 1210: Worksheet #9
Name:
Winter 2017
A#:
Section:
1. (6) Compute the limit and explain why L’Hôpital’s rule does not apply.
x + sin(x2 )
x→−∞
ex − 2x
lim
L’Hôpitals rule does not apply because:
2. (6) Compute the following limits.
x − sin(x)
x→0
x3
(a) lim
1 ln(1 − x)
(b) lim
+
x→0 x
x2
!
v9.2
3. (4) Find the linearisation L(x) of f (x) = tan−1 (x) centred at 1.
4. (4) Use linearisation to find a reasonable approximation of ln(0.95).
Bonus. (5) Let a, L be a real numbers and let f be a continuous function such that lim f 0 (x) = L.
x→a
Prove that f is differentiable at a and that f 0 (a) = L.
v9.2
Math 1210: Worksheet #10
Name:
Winter 2017
A#:
Section:
1. (3) Set up the definite integral as a limit of left-endpoint Riemann sums. Do not evaluate.
Z 5
2
3
e1/t dt
2. (3) Identify the limit as a definite integral. Do not evaluate.
lim
N →∞
N X
i=1
2i 2
2i
e(1+ N ) + 2 1 +
N
2
N
3. (5) Solve the initial value problem: y 00 = sin(2x), y 0 (0) = 0, y(0) = 0.
v10.1
4. (9) Compute the indefinite integrals.
(a)
Z
x2 + 4x + 5
√
dx
x
(b)
Z
1
dx
x2 + 4x + 5
(c)
Z
1
dt
(3t + 1)42
v10.1
Math 1210: Worksheet #10
Winter 2017
Name:
A#:
Section:
1. (3) Set up the definite integral as a limit of left-endpoint Riemann sums. Do not evaluate.
Z 3
1
sin(t2 + 1)dt
2. (3) Identify the limit as a definite integral. Do not evaluate.
lim
N →∞
N X
i=1
tan 3 +
4i
4i
+ cos 3 +
N
N
4
N
3. (5) Solve the initial value problem: y 00 = e2x , y 0 (0) = 0, y(0) = 0.
v10.2
4. (9) Compute the indefinite integrals.
(a)
Z
x2 + 2
dx
x
(b)
Z
x2
(c)
Z
√
1
dx
+2
1
dt
1 − 3t2
v10.2
Math 1210: Worksheet #11
Name:
Winter 2017
A#:
Section:
1. (5) Compute the derivative.
d Z et x2
e dx
dt t2
2
2. (15) Compute the following definite and indefinite integrals. Choose any three for full
marks. Indicate your choice clearly.
(a)
Z
(b)
Z
sin3 (x) cos4 (x)dx
x2
1
dx
+ 2x + 3
v11.1
(c)
Z
(d)
Z ln(3) 2x
e
(e)
Z 1
ln(3)
2
0
0
0
ex
dx
1 + e2x
+1
dx
ex
√
(x − 1) x + 1 dx
v11.1
Math 1210: Worksheet #11
Name:
Winter 2017
A#:
Section:
1. (5) Compute the derivative.
d Z t3 +t
sin(x3 )dx
dt ln(t)
2. (15) Compute the following definite and indefinite integrals. Choose any three for full
marks. Indicate your choice clearly.
(a)
Z
x2
(b)
Z
√
1
dx
+ 4x + 6
t3
dt
2 − t2
v11.2
(c)
Z − ln(2)
2
(d)
Z 1
y2
dy
(y + 1)3
(e)
Z 0
ex − e−x
dx
e2x
− ln(2)
0
−1
ex
√
dx
1 − e2x
v11.2