calculus 1, part 1

1. C H A P TE R 0
A P R E V IE W O F
C ALC ULUS

2. C O N TE N TS
 1. T h e A r e a P r o b le m
 2 . The Ta ng e n t P r o b le m
 3 . V e lo c it y
 4 . T h e L im it of a
S e que nc e
 5 . The S u m o f a S e r ie s

3. 1. THE AREA PROBLEM
 The area of a polygone can be
defined by subdividing it into
F
triangles A
E
B
C D
Area (ABCDEF)=Area(ABF)+Area(BCF)+Area(CDF)+Area(DEF)

4. 1. THE AREA PROBLEM
 The area of a curved figure can be defined
as limit of areas of incribed and
circonscribed polygones
A1 A2 A3 A4 A5
Area(Circle) = lim An
n →∞

5. How to compute the area under the
parabola y = x2 ?

6. It is approximated by the sum of the
areas of circumscribed rectangles

7. the approximation can be refined by
taking narrower circumscribed
rectangles

8. Finally the area can be computed as the limit
of the sum of the areas of circumscribed
rectangles
This is the idea of Integral Calculus

9. 2 . TH E TA N G E N T
PROB LEM
t
P
The tangent t to the curve y=f(x) at point
P is determined by its slope m

10. We can approximate m by the slope mP Q of a
secant P Q , where Q is a nearby point
t
P (a,f(a)) Q (x,f(x))
f ( x) − f (a)
mPQ =
x−a

11. f ( x ) − f (a )
m = lim mPQ = lim
Q→P Q→P x−a
This is the idea of Differential Calculus

12. 3.
V E L O C IT Y
The distances (in feet) traveled by a car moving
along a straight road measured at 1-second
intervals are given by:
t = time elapsed 0 1 2 3 4 5
(s)
d = distance (ft) 0 2 10 25 43 78
The average velocity in the time interval 2 ≤ t ≤ 4
is:
43 − 10
v= = 16.5 ft / s
4−2

13. The average velocity in the time interval 2 ≤ t ≤ 3
is:
25 − 10
va = = 15 ft / s
3− 2
The distances traveled by the car measured at 0.1-
second intervals are given by:
t 2.0 2.1 2.2 2.3 2.4 2.5
d 10.0 11.0 12.16 13.45 14.96 16.80
0 2
The average velocity in the time interval 2 ≤ t ≤ 2.5
is:
16.80 − 10.00
va = = 13.6 ft / s
2.5 − 2

14. The average velocity computed in various interval
is:
t [2 3] [2 2.5] [2 2.4] [2 2.3] [2 2.2] [2 2.1]
va 15.0 13.6 12.4 11.5 10.8 10.2
The instantaneous velocity at t = 2 is the limit of the
average velocity over smaller and smaller interval [2
t]:
f (t ) − f (2)
v = lim
t →2 t −2
where f(t) is the distance (in feet) traveled
after t seconds

15. 4 . L IM IT O F A
S EQUENC E
The real number π is the limit of the sequence:
a1=3.1, a2=3.14, a3=3.141, a4=3.1415, a5=3.14159,
…
π = lim an
n →∞
1
lim = 0
n →∞ n

16. 5 . TH E S U M O F A
S E R IE S
The n-digit decimal representation of a rational
number 0.d1 d2 d3 … dn is the finite sum:
d1 d2 d3 dn
0.d1d 2 d 3 ...d n = + 2 + 3 +... + n
10 10 10 10
The infinite decimal representation of a real
number 0.d1 d2 d3 … dn … is the infinite sum:
d1 d2 d3
0.d1d 2 d 3 ... = + 2 + 3 +...
10 10 10

17. Therefore the infinite decimal representation 0.333 …
is:
1 3 3 3
= 0.333... = + 2 + 3 +...
3 10 10 10
Similarly we can compute the infinite
sum:
1 1 1 1
+ + + ... + n + ... As the limit of the sequence:
2 4 8 2
1 1 1 1 1 1
s1 = , s2 = + ,..., s10 = + + ... + = 0.99902344,...,
2 2 4 2 4 1024
1 1 1 1 1 1 1
s16 = + + ... + 16 = 0.99998474,..., sn = + + + ... + n ,...
2 4 10 2 4 8 2
1 1 1 1
+ + +... + n +... = lim sn =1
2 4 8 2 n→∞