Write-ups
of the problems below are due Monday Nov. 5
Monday
Oct 30
Topics: 6.4
Solve, write-up and turn in:
1. Consider the function f(x)=2x on the interval [0,b].
Set up a Riemann sum to calculate the area under f(x)=2x for x in [0,b]
as follows.
A. Consider a partition of [0,
b] made up of 4 intervals of equal lengths. What are x0,x1,x2,x3,x4?
a) On each interval [xk,x
k+1] take the intermediate points to be the left points
(so x*k=xk).
Sketch the four rectangles and write the Riemann sum (do not simplify!)
Is this Riemann sum less, more, or equal to the actual area? Justify
your answer by looking at the picture.
b) Same questions when you
choose x*k to be the right points (so x*k=x
k+1).
c) Same questions when you choose
x*k to be the mid-points (so x*k=(xk+x
k+1)/2).
B. Consider a partition of [0,
b] made up of n intervals of equal lengths. Write x0,x1,x2,...xk,...xn.
On each interval take the
intermediate points to be the left points (so x*k=xk).
(i) Write the Riemann sum.
(ii) Calculate the sum.
(iii) Calculate the limit as
n-> infinity and show that this limit gives the area indeed.
(There was an y=ax floating around...the function should be y=2x.)
Tuesday
Oct 31
Topics: 6.5,
6.6
Solve, write-up and turn in: 2. Consider
the function f(x)=a x^2 on the interval [0,b] (the one we did in class
today).
Write a Riemann sum associated to n equal intervals, and with
intermediate points being midpoints.
Calculate the limit as n goes to infinity of this Riemann sum.
What should this limit be? Explain!
Sec. 6.6: 3, 5, 9, 11, 13, 16, 20, 37, 42
Wednesday
Nov. 1
Topics: 6.7,
7.2
Solve, write-up and turn
in: Sec. 6.7: 1a, d (Caution: sketch the graph and note
that part of the function is negative,
so the geometrical area of that part is "minus the definite integral"),
3, 7, 9, 11, 13, 15a, 16 a,b
3. Calculate the area between the graphs of y=sin x and
y=cos x for x in [0,pi].
Caution: sketch the two graph and note that cos x is bigger on one
interval, and sin x is bigger on another interval.
Thursday
Nov. 2
Topics: 7.2,
7.3
Solve, write-up and turn
in: Sec. 7.2: 20, 25, 29a, 31, Bonus: 33
Sec. 6.7: 14
Sec. 7.3: 1 c, f, 2, 11
Friday
Nov. 3
Topics: 7.3
Solve, write-up and turn
in: Sec. 7.3: 7, 9, 15, 22, 25