Ordinary differential equations
Professor Kwok Wing CHOW
Mechanical Engineering, HKU
Motivation There are many situations in
science and engineering where a
quantitative relation exists between a
function and its rate of change. Examples
are
(a) Populatio
MATH 1851B - Quiz 1 Outline Solution
1. (25%) Find
lim sin xtan x .
x0+
Write down the names of those theorems you have used.
Solution. Let y = sin xtan x . Then log y = tan x log sin x = log sin x , which tends to the
cot x
indeterminate form / as x 0+ .
MATH 1851A - Quiz 1 Outline Solution
1. (25%) Find
1
x2 sin x
.
x0 log(1 + x)
Write down the names of those theorems you have used.
lim
+
Solution. One sees that
1
x2 sin x
x2
,
ln(1 + x)
ln(1 + x)
for any x > 0. Using the LHpitals rule, the limit
o
[0]
x
Chapter 6.
Applications of Denite Integrals
The number in [ ] refers to the page number of our textbook
6.1 Volumes Using Cross-Sections
Denition [289] The volume of a solid of integrable cross-sectional area A(x) from x = a
to x = b is the integral of A
Chapter 8
Techniques of Integration
Indenite Integral - Denition [184, 189] A function F is an antiderivative of f on an
interval I if F (x) = f (x) for all x in I . The collection of all antiderivatives of f is called
the indenite integral of f with resp
MATH1851 ODEs Tutorial Ex
dy
dt
2y = 4 t.
7
t
Answer: y = 4 + 2 + Ce2t
1.
Solve the following ordinary dierential equation
2.
dy
Solve the following Bernoullis equation 2y 3 dx = y 4 e3x .
2x
1
5e
Answer: y = ( e5x +C ) 3
3.
[From lecture notes] Use sepa
Chapter 5
Integration
The number in [ ] refers to the page number of our textbook
5.1 Area under a curve
Eg. 1 [228] - using upper and lower sums to estimate the area under a curve. See Table 5.1
[229]
Eg. 2 [229] - distance travelled by a projectile.
Re
Q1. Calculate the upper and lower Riemann
sum for f (x) = 1 on [1, 5] using 4 equal inx
tervals. Compare these values with the actual
area under graph.
Ans. The interval is divided into 4:
[1, 2], [2, 3], [3, 4], [4, 5]
The function f is decreasing on [1,
Q2. Find:
x3(x + 1)dx
4+ t
dt .
t3
,
Ans. Rewrite the rst integral as
(x2 + x3)dx ,
then nd the antiderivative of x2 + x3 by
asking yourself: by dierentiating which function will give you x2 + x3? And the answer
x1 + x2 , so
is 1
2
1
x 2
3 (x + 1)dx = x
x
Q2. Sketch y =
2
x 3 (x
5).
Ans. It is clear that the Domain can be R.
1
5
* y = 3 x 3 (x 2). So y > 0 if x > 2 or x < 0;
y < 0 if 0 < x < 2; y = 0 when x = 2. Hence,
y increasing on (, 0) and (2, );
y decreasing on (0, 2);
y attained local min. at x = 2
Q1. If functions f (x) and g (x) are continuous
for 0 x 1, could f (x)/g (x) possibly be discontinuous at a point of [0, 1]? Give reasons
for your answer.
Critic. It is a well known fact that if f and g are
continuous at c, then f /g will be continuous at
Math1851, Tutorial 5 (for 22/10-26/10)
FLT
Please work on the following 8 problems before your tutorial session.
1. (Ex5.1: Q3) Calculate the upper and lower Riemann sum for the func1
tion dened by f (x) = x between 1 and 5, suppose you are dividing 4
equ
Math1851, Tutorial 3 (for 8/10-12/10)
FLT
Please go through all 10 problems before your tutorial session.
1. Read Example 8 on page 162: Sketch the graph of f (x) =
(x+1)2
.
1+x2
To sketch a graph (end product), one needs to understand the function
f . Fo
Math1851, Tutorial 1 (for 24/9-28/9)
FLT
Please work on the following 10 problems before your tutorial session.
1. (Ex1.1: #6) Find the average rate of change of the function dened by
P () = 3 42 + 5 on the interval [1, 2].
2. (Ex1.1: #12) Find the tangen
MATH1851 Laplace Transforms Tutorial Ex
1.
Consider the function f (t) = tn , where n is a positive integer.
(a)
Determine the Laplace transform of f (t) from denition.
Answer:
(b)
n!
sn+1
Using the rst shift property, nd L(eat f (t).
Answer:
n!
(sa)n+1
2