- In the chapter, upper and lower sums were considered. Suppose g is an increasing
function and you are considering upper and lower sums for approximating ∫
Show that when you add in a point to the partition, the upper sum which results
is no larger but the lower sum is no smaller.
- Let f = 1 +
x2 for x ∈ and let
. Find U
x and for F =
- Let P = and
x. Find upper and lower sums for the
function f =
using this partition. What does this tell you about ln
- If f ∈ R with
x and f is changed at finitely many points, show
the new function is also in R
. Is this still true for the general case where
F is only assumed to be an increasing function? Explain.
- In the case where F =
x, 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 in 9.6, ∫
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
- The function F
≡ gives the greatest integer less than or equal to
F = 0
,F = 5
,F = 5
, etc. If F =
as just described, find
More generally, find ∫
dF where f is a continuous function.
- Suppose f is a bounded function on and for each
ε > 0,∫
dx exists. Can
you conclude ∫
- A differentiable function f defined on satisfies the following conditions.
Find f and sketch its graph.
- Does there exist a function which has two continuous derivatives but the third
derivative fails to exist at any point? If so, give an example. If not, explain
- Suppose f is a continuous function on and
where F is a strictly increasing integrator function. Show that then f = 0 for all
F is not strictly increasing, is the result still true?
- Suppose f is a continuous function and
for n = 0,1,2,3
. Show using Problem 10 that f = 0 for all
x. Hint: You might
use the Weierstrass approximation theorem.
- 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 with respect to the integrator on the interval ?
- Recall that for a power series,
you could differentiate term by term on the interval of convergence. Show that if the
radius of convergence of the above series is r > 0 and if
In other words, you can integrate term by term.
- Find ∑
- 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
. Show using the fundamental theorem of calculus and the product
rule the integration by parts formula. Also explain why all the terms make
Now use this to find a series which converges to arctan =
For which values of x will your series converge? For which values of x does the above
description of arctan in terms of an integral make sense? Does this help to show the
inferiority of power series?
- Define F
dt. Of course F = arctan
as mentioned above but just
consider this function in terms of the integral. Sketch the graph of
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 equals
- In Problem 19 let the limit defined there be denoted by π∕2 and define T
. Show T′ = 1 +
2 and T = 0
. As part of this, you must
explain why T′ exists. For
x ∈ let
≡ 1∕ with
C = 0
, define S by
. Show both S and
. Find the
appropriate way to define S and
C on all of
in order that these functions
will be sin
and then extend to make the result periodic of period 2
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
Now use the binomial theorem to find a power series for arcsin
- The initial value problem from ordinary differential equations is of the form
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∕
= xq−1. 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 Stieltjes integrable functions on with respect to
an increasing function. Verify Holder’s inequality.
Hint: Do the following. Let A =
and use the wonderful inequality of Problem 23.
- Let F =
dt. Find F′
- Let F =
dt. Sketch a graph of F and explain why it looks the way it
- Let a and b be positive numbers and consider the function
Show that F is a constant.
- Solve the following initial value problem from ordinary differential equations which is to
find a function y such that
- If F,G ∈∫
dx for all x ∈ ℝ, show F =
C for some constant, C. Use this
to give a different proof of the fundamental theorem of calculus which has for its
dt = G
− G where
- 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
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.
- The Riemann integral is only defined for bounded functions on bounded intervals. When
f is Riemann integrable on for each
R > a define an “improper” integral as
whenever this limit exists. Show
exists. Here the integrand is defined to equal 1 when x = 0, not that this
- If you did not read the material on the Gamma function, here is an exercise which
introduces it. The Gamma function is defined for x > 0 by
Give a meaning to the above improper integral and show it exists. Also show
and for n a positive integer, Γ =
n!. Hint: The hard part is showing the integral
exists. To do this, first show that if f is an increasing function which is bounded
above, then lim
x→∞f must exist and equal sup
is an increasing function of R which is bounded above.
- 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
, verify 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.
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,
values of a function
f defined on this interval are fi = f