- Define a “left sum” as
and a “right sum”,
Also suppose that all partitions have the property that xk −xk−1 equals a constant,
∕n so the points in the partition are equally spaced, and define the integral
to be the number these right and left sums get close to as n gets larger and
larger. Show that for f given as 1 on rational numbers and 0 on irrational
dt = x if x is rational and ∫
dt = 0 if x is irrational.
It turns out that the correct answer should always equal zero for that
function, regardless of whether x is rational. This illustrates why this
method of defining the integral in terms of left and right sums is total
nonsense. Show that even though this is the case, it makes no difference if f is
- Suppose f is a bounded function on and for each
ε > 0,∫
Can you conclude ∫
- The improper integrals discussed in the chapter had to do with an infinite interval
of integration. Another kind of improper integral is considered when you try to
integrate an unbounded function. Here is an example:
dx for various values of α. Consider what happens when α < 1 and
when α ≥ 1.
- A differentiable function f defined on satisfies the following conditions.
Find f and sketch its graph.
- Suppose f is a continuous function on and
Show that then f = 0 for all
- Here is a function:
Show this function has a derivative at every point of ℝ. Does it make any sense to
Is f Riemann integrable on the interval ?
- Let f be Riemann integrable on
. Show directly that x →∫
continuous. Hint: It is always assumed that Riemann integrable functions are
- Suppose f,g are two functions which are continuous with continuous derivatives on
. Show using the fundamental theorem of calculus and the product rule
the integration by parts formula. Also explain why all the terms make
- Define F
dt. Of course F = arctan
as mentioned above but
just consider this function in terms of the integral. Sketch the graph of
F using only
its definition as an integral. Show there exists a constant M such that
−M ≤ F
≤ M. Next explain why limx→∞F exists and show this limit
- In Problem 10 let the limit defined there be denoted by π∕2 and define
≡ F−1 for
. Show T′ = 1 +
2 and T = 0
As part of this, you must explain why T′ exists. For
x ∈ let
≡ 1∕ with
C = 0 and on
, define S by
. Show both S and
C are differentiable on
. Find the appropriate way to define S and
C on all of
in order that these functions will be sin
then extend to make the result periodic of period 2
π on all of ℝ. Note this is a
way to define the trig. functions which is independent of plane geometry
and also does not use power series. See the book by Hardy,  for this
- The initial value problem from ordinary differential equations is of the
Suppose f is a continuous function of y. Show that a function t → y solves the
above initial value problem if and only if
- Let p,q > 1 and satisfy
Consider the function x = tp−1. Then solving for t, you get t = x1∕
Explain this. Now let a,b ≥ 0. Sketch a picture to show why
Now do the integrals to obtain a very important inequality
When will equality hold in this inequality?
- Suppose f,g are two Riemann integrable functions on . Verify Holder’s
Hint: Do the following. Let A =
and use the wonderful inequality of Problem 13.
- If F,G ∈∫
dx for all x ∈ ℝ, show F =
C for some constant, C. Use
this to give a proof of the fundamental theorem of calculus which has for its
dt = G
. Use the version of the
fundamental theorem of calculus which says that
′ = f for
- Suppose f is continuous on . Show there exists
c ∈ such that
Hint: You might consider the function F
dt and use the mean
value theorem for derivatives and the fundamental theorem of calculus.
- Suppose f and g are continuous functions on and that
Show there exists c ∈ such that
Hint: Define F
dt and let G
dt. Then use the
Cauchy mean value theorem on these two functions.
- Consider the function
Is f Riemann integrable on ? Explain why or why not.
- When f is Riemann integrable on for each
R > a the “improper” integral is
defined as follows.
whenever this limit exists. Show
exists. Here the integrand is defined to equal 1 when x = 0, not that this
- The most important of all differential equations is the first order linear equation,
y′ + p
y = f where
p,f are continuous. Show the solution to the initial value
problem consisting of this equation and the initial condition, y =
where P =
ds. Give conditions under which everything is correct.
Hint: You use the integrating factor approach. Multiply both sides by eP
the left side equals
and then take the integral, ∫
at of both sides.
- Suppose f is a continuous function which is not equal to zero on
Hint: First change the variables to obtain the integral equals
Next show by another change of variables that this integral equals
Thus the sum of these equals b.
- Let there be three equally spaced points, xi−1,xi−1 + h ≡ xi, and xi + 2h ≡ xi+1.
Suppose also a function f, has the value fi−1 at x, fi at x + h, and fi+1 at x + 2h.
Then consider Check that this is a second degree polynomial which equals the values fi−1,fi, and
fi+1 at the points xi−1,xi, and xi+1 respectively. The function gi is an
approximation to the function f on the interval
is an approximation to ∫
dx. Show ∫
Now suppose n is even and is a partition of the interval,
and the values of a function
f defined on this interval are fi = f
Adding these approximations for the integral of f on the succession of
show that an approximation to ∫
This is called Simpson’s rule. Use Simpson’s rule to compute an approximation to
dt letting n = 4.
- Suppose x0 ∈ and that
f is a function which has n + 1 continuous derivatives
on this interval. Consider the following.
|| = f +
| || = f +
x0x + ∫
| || = f +
Explain the above steps and continue the process to eventually obtain Taylor’s
where n! ≡ n
⋅ 2 ⋅ 1 if n ≥ 1 and 0! ≡ 1.
- In the above Taylor’s formula, use Problem 17 on Page 610 to obtain the existence
of some z between x0 and x such that
Hint: You might consider two cases, the case when x > x0 and the case when
x < x0.
- There is a general procedure for constructing methods of approximate integration
like the trapezoid rule and Simpson’s rule. Consider and divide this interval
n pieces using a uniform partition, where
xi − xi−1 = 1∕n for
each i. The approximate integration scheme for a function f, will be of the
where fi = f and the constants,
ci are chosen in such a way that the above
sum gives the exact answer for ∫
dx where f = 1
this has been done, change variables to write where fi = f
. Consider the case where n = 1. It is necessary to
find constants c0 and c1 such that
Show that c0 = c1 = 1∕2, and that this yields the trapezoid rule. Next take n = 2
and show the above procedure yields Simpson’s rule. Show also that if this
integration scheme is applied to any polynomial of degree 3 the result will be exact.
whenever f is a polynomial of degree three. Show that if
fi are the values of f
, and b with f1 = f
, it follows that the above formula gives