Math 1B: Discussion Exercises
GSI: Theo Johnson-Freyd
http://math.berkeley.edu/~theojf/09Summer1B/
Find two or three classmates and a few feet of chalkboard. Introduce yourself to your new
friends, and write all of your names at the top of the chalkboard. As a group, try your hand at
the following exercises. Be sure to discuss how to solve the exercises — how you get the solution
is much more important than whether you get the solution. If as a group you agree that you all
understand a certain type of exercise, move on to later problems. You are not expected to solve all
the exercises: some are very hard.
Many of the exercises are from Single Variable Calculus: Early Transcendentals for UC Berkeley
by James Stewart; these are marked with an §. Others are my own, are from the mathematical
folklore, or are independently marked.
Here’s a hint: drawing pictures — e.g. sketching graphs of functions — will always make the
problem easier.
Approximate Integration
To approximate Rb
af(x)dx, let ∆x= (b−a)/n,xi=a+i∆x, and ¯xi= (xi−1+xi)/2. Then
define the following approximations (we use different number from Stewart for Simpson’s Rule Sn):
Ln= (f(x0) + · ·· +f(xn−1)) ∆x Rn= (f(x1) + · ·· +f(xn)) ∆x
Tn= (f(x0)+2f(x1) + · ·· + 2f(xn−1) + f(xn)) ∆x
2Mn= (f(¯x1) + · ·· +f(¯xn)) ∆x
Sn= (f(x0)+4f(¯x1)+2f(x1)+4f(¯x2)+2f(x2) + · ·· + 2f(xn−1)+4f(¯xn) + f(xn)) ∆x
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These have the following errors:
|EL| ≤ sup
x∈[a,b]
f0(x)
(b−a)2
2n|ER| ≤ sup
x∈[a,b]
f0(x)
(b−a)2
2n
|ET| ≤ sup
x∈[a,b]
f00(x)
(b−a)3
12n2|EM| ≤ sup
x∈[a,b]
f00(x)
(b−a)3
24n2
|ES| ≤ sup
x∈[a,b]
f(4)(x)
(b−a)5
2780n4
The word “sup” is short for “supremum” — the symbol “supx∈[a,b]g(x)” means “the largest value
of g(x) as xranges over [a, b]”. In practice, it suffices to replace the suprema with some easy-to-
compute numbers which are even bigger.
1. Explain each of the above approximation techniques when n= 1.
2. Let f(x) be a positive increasing function with negative second derivative on [a, b]. Place the
following five numbers in increasing order: Ln,Rn,Tn,Mn, and Rb
af(x)dx.
3. If you were to evaluate each of the following integrals using each of the Midpoint, Trapezoid,
and Simpson Rules with 5 subintervals, what would be your expected error? How many
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