CALCULUS I
Dr. Nguyen Ngoc Hai
DEPARTMENT OF MATHEMATICS
INTERNATIONAL UNIVERSITY, VNU-HCM
April 19, 2010
Dr. Nguyen Ngoc Hai
CALCULUS I
References
Main textbook:
J. Steward, Calculus. Concepts and Contexts, 5th ed., Thomson
Learning, 2001.
Other textbook
Chapter 4. THE INTEGRAL
4.1
THE AREA PROBLEM
Problem:
Find the area of the region S under the curve y = f (x ) over
an interval a x b, where f (x ) 0 and f is continuous on
[a, b ].
This means that S is bounded by the graph of a continuous
function f , th
Chapter 2 DIFFERENTIATION
Contents
[1.] Tangent Lines and Their Slopes
[2.] Derivatives
[3.] Dierentiation Rules
[4.] Rates of Change in Natural and Social Sciences
[5.] Derivatives of Trigonometric Functions
[6.] Chain Rule and Implicit Dierentiation
[7.
Chapter 3. APPLICATIONS
OF DIFFERENTIATION
Contents
1. Related Rates
2. Extreme Values
3. The Mean Value Theorem. Shapes of Curves
4. Sketching the Graph of a Function
5. LHpitals Rule
o
6. Optimization Problems
7. Newtons Method
8. Antiderivatives and In
Chapter 5. APPLICATIONS OF INTEGRATION
5.1
AREAS BETWEEN CURVES
5.1.1
AREA BETWEEN TWO CURVES
y = f (x ) AND y = g (x )
If f (x ) g (x ) 0 for x [a, b ], then the graph of f lies above
the graph of g , and the region between the two graphs is
b
b
f (x )dx
1.
lim x 3 f ( x )=3
means that as x approaches 3,
f (x)
gets closer to 3. It can
get infinitely close but not touch. So it can be
2.999999999999999999999999999999999 and be ok but it cannot be 3.
2.
a. C(x)= 0.5*50/100-50= 0.5
C(x)= 0.5*60/100-60= .75
C(
1)
a. no limit
b. I am going to say no limit. I used 3 different calculators and got 3 different graphs.
2)
a. 2
b. 3
c. does not exist
d. does not exist
e. infinity
f. infinity
g. 1
x2
3) B(x)= x 2 +4
a. month 1 is 0.2 million dollars, month 2 is 0.5 mil