Write-ups due - extended to Tuesday Oct. 3:

Monday Sept 25

Topics discussed in lecture today: Sec. 2.3.2 in Bounds of sets of numbers
and Appendix B.2 (p.849, 850, top of 851)
(see again the top of the page with exercises).

Solve, write up and turn in: Exercises above (p.6) 2.3.2,  2.3.3 d,e,f
Prove by induction formulas (4) and (5) page 851

Tuesday Sept 26

Topics discussed in lecture today: induction,
arithmetic and geometric progressions
(web resource - see Ch.11)

Useful formula: the sum of the first n terms of an arithmetic progression with common difference d is
a1 +a2 +...+ an = n a1 + d n(n-1)/2.

Solve: Show, using induction, that the general term of a geometric progression b1,b2,b3,... of common ratio r is
bn = b1 r^(n-1)      Notation: b^n denotes b to the power n

Solve, write up and turn in:
B.1 problems 1a,b,c, 4b, 8

and also 1. Which of the sums in problem 1a,b,c [are sums of] arithmetic progressions? Find the first term and the common difference,
and calculate the sums using the formula for the sum of the first n terms of an arithmetic progression.
2. Calculate the sums (note that the terms are geometric progression!):
a. 1+3+3^2 +3^3+...+3^n
b. 1+r+r^2+...+r^n
c. x+x^2+x^3+...+x^n
d. 1-s+s^2-s^3+...+(-1)^n s^(n)

Wednesday Sept 27
Today's topics: calculation of (some type of) sums.

Solve, write up and turn in: from the textbook, Sec. B1 (page 854) 1c,d,e (you may use any method)

also: 1. Calculate   3+6+9+...+(3n).
2.  Calculate   1/2 -1/4 +1/8 - 1/16 +...+ (-1)^(n+1)/2^n   . Is the general term correctly written?
3.  Calculate   (2n+1) + (2n+2) + (2n+3) +...+ (4n).

Thursday Sept 28

Topcs: sums, review the absolute value, start Sec. 2.5
Reminder: the distance between the numbers a and b plotted on the number line is |a-b|.
Triangle's inequality:
|a+b|<|a|+|b|

Solve: if you need more practice with sums, try the following:
calculate the sum in problem B1 4a; calculate  1-1/2+1/4-1/8+...+(-1)^n/2^n;
calculate 1+2+3+...+(8n+3);   calculate 101+102+103+...+n (where n>100, for the sum to make sense).

Solve, write up and turn in: Consider the function f(x)=(4x^2-4)/(x-1).
1.
What is the domain of this function?
2.
Plot the function. (Note that for x not equal to 1 the fraction can be simplified, yielding 4x+4.)

By inspecting the graph, it appears that f(x) is as close to 8 as we wish provided that x is close enough to 1 (but not 1).
Let us check this:
3.
Show that  |f(x)-8| < .1 (so: the distance between f(x) and 8 is smaller than .1) provided that
|x-1| < .1 /4 and x not 1(so: the distance between x and 1 is smaller than .1/4, but not 0).
4.
|f(x)-8|< .00001 provided that x is how close to 1(but not 1)?
5.
Let us denote by epsilon a very, very small positive number (think like .00000000001, or like 10^(-10,000), or maybe even smaller...)
Then |f(x)-8|<epsilon provided that x is how close to 1 (but not 1)?

Bonus problem (extra 5 points): Prove the triangle inequality: |a+b|<|a|+|b|

Math tutor locations and hours

Friday Sept 28

Topcs: 2.5, A.2  Theorems 1, 2(proved), 3(proved), 4(proved i), 5.
Solve, write up and turn in: 1. Use the epsilon-delta definition of limits to show that if a function f(x) has the limit L as x->a
(write for short  ' f(x)->L as x->a ') and if c is a constant, then     cf(x)->cL as x->a.  (This is the last property we discussed in class today.)
2. Use the properties listed in class (see A.2 Theorems 1,2,3,4,5) to show that the following are true as as x->a :
a) x^2 -> a^2
b)
x^3-> a^3
c)
x^n-> a^n  (use induction!)
3.
Use the properties listed in class and problem 2. to argue* that if P(x) and Q(x) are polynomials, then
a)  P(x) -> P(a)   as x->a
b) P(x)/Q(x) -> P(a)/Q(a)  as x->a  if Q(a) is not zero.
*In this problem 'argue' means that you can use '... ' and you are spared a proof by induction (unlike 2.c, where the property is clear using '...', yet you are asked to
use induction nevertheless).
Bonus problem: (extra 3 points) Use induction to show that   |a1+a2+...+an| <| a1|+|a2|...+|an|.

Write-ups of the problems above: due date extended to Tuesday Oct. 3

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