MATH1151 Student Support
Revision Questions
1.
(a)
Z
ln 3
0
x
sinh
x dx
(b) Show that cosh
2
x

sinh
2
x
= 1 and deduce that tanh
x
is an increasing function.
2. Determine each of the following limits. Formally prove your answers to (a) and (b).
(a)
lim
x
→
1
+
1
√
x

1
(b) lim
x
→∞
x
2
+ 1
x
2
(c) lim
x
→
0
x
log(1 +
x
)
(
e
x

1)
2
(d) lim
x
→
1
Z
x
1
cos(
t
2
)
dt
sin(
πx
)
(e)
lim
x
→
2
+
x
2

x

2
√
x

2
3. Suppose that
f
(
x
) =
1
2
(
x
+ 1)
for

1
≤
x <
1
,
x
2

4
x
+ 4
for 1
≤
x <
2
,

x
2
+ 4
x

4
for 2
≤
x <
3
,
f
(
x
+ 4)
for all real
x.
Is
f
continuous at
x
= 1? Is
f
continuous at
x
=

1? Is
f
differentiable at
x
= 2? Give reasons.
4. Sketch the polar curve
r
=

sin
θ

where 0
≤
θ
≤
2
π
and determine the cartesian equation(s) of the curve.
5.
(a) Divide the interval 1
≤
x
≤
2 into
n
equal parts, and show that
n
X
j
=1
1
n
+
j
<
Z
2
1
1
x
dx <
n

1
X
j
=0
1
n
+
j
.
(b) Find the difference between the upper and lower sums.
Use this to show that the error in the approximation ln 2
≈
10
5
X
j
=1
1
10
5
+
j
is less than 5
×
10

6
.
6.
(a) Show that
Z
∞
0
1
x
+
√
x
2
+ 1
φ
dx
= 1, where
φ
=
1 +
√
5
2
.
(b) Prove that the improper integral
Z
∞
1
√
x
1 + 5
x
2
dx
converges and give a numerical upper bound for its value.
(c) Evaluate the improper integral
Z
2
1
2
x

3
√
x
2

1
dx
after first writing it explicitly as a limit.
7. Show that
e
x
≥
1 +
x
for all real numbers
x
.
Then, using the fact that
e
x
= 1 +
Z
x
0
e
t
dt
, show that
e
x
≥
1 +
x
+
x
2
2
for all positive real numbers
x
.
8.
(a) Suppose
a >

1 and
b >

1. Use Leibniz’s rule to show that
Z
1
0
x
a

1
ln
x
dx
= ln(
a
+ 1).
Hence, or otherwise, evaluate
Z
1
0
x
a

x
b
ln
x
dx
(b) Let
x
(
t
) =
1
2
Z
t
0
sinh[2(
t

s
)]
f
(
s
)
ds
where
f
is a continuous function. Show that
d
2
x
dt
2

4
x
=
f
(
t
).
9. The error term involved in using Simpson’s Rule can be written as
Ef
=

f
(4)
(
c
)(
b

a
)
5
2880
n
4
where
c
∈
(
a, b
).
Estimate the number of subintervals needed to evaluate
Z
3
1
dx
x
3
with Simpson’s Rule and error less than 10

6
.
10. Suppose
w
=
x
+ sin(
yz
) and
x
= 2
s
+ 1
, y
=
s

t, z
=
te
s
.
Determine
∂w
∂s
at (
s, t
) = (0
,
1).
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MATH1151 Student Support
Revision Questions
11. Consider the plane
π
that passes through the points
A
(1
,
0
,

1)
T
, B
(3
,

1
,
0)
T
and
C
(2
,
1
,
1)
T
.
Find the angle
ABC
, the area of the triangle
ABC
, the equation of
π
in parametric vector, cartesian and
pointnormal forms, and the point(s) of intersection, if any, of
π
with the line
x

1
3
=
y
+ 2
1
=
z
6
.
12. Find the shortest distance between:
(a) the point (8
,
1
,

5)
T
and the line
˜
x
= (1
,
2
,

1)
T
+
λ
(3
,
1
,
0)
T
;
(b) the point (6
,

5
,
8)
T
and the plane
x

y
+ 3
z
= 2;
(c) the lines
˜
x
= (1
,

3
,
9)
T
+
λ
(3
,
2
,
5)
T
and
˜
y
= (5
,
0
,
4)
T
+
μ
(2
,
1
,
3)
T
.
13. Use the method of least squares to determine the line of best fit: (

2
,
3), (1
,

2), (4
,

7), (5
,
2).
14. Show that the matrix
A
=
1
2
5
3
2

1
2
1
0
is invertible, and find
A

1
.
Verify that
AA

1
=
I
and
A

1
A
=
I
.
15. Determine the values of
a
and
b
such that the system of linear equations:

x
+ 4
y
=

5
and
3
x
+
ay
=
b
has (a) infinitely many solutions; (b) no solution; (c) a unique solution.
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 Three '11
 Emoticon, Hyperbolic function, MATH1151 Student

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