- 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 ∫
_{a}^{b}fdg. 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 + x
^{2}for x ∈and let P =. Find Uand Lfor F= x and for F= x^{3}. - Let P = and F= x. Find upper and lower sums for the function f=using this partition. What does this tell you about ln?
- If f ∈ Rwith F= 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 x

_{k}− x_{k−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, ∫_{0}^{x}fdt = x if x is rational and ∫_{0}^{x}fdt = 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 continuous. - The function F≡gives the greatest integer less than or equal to x. Thus F= 0 ,F= 5 ,F= 5 , etc. If F=as just described, find ∫
_{0}^{10}xdF. More generally, find ∫_{0}^{n}fdF where f is a continuous function. - Suppose f is a bounded function on and for each ε > 0,∫
_{ε}^{‘1}fdx exists. Can you conclude ∫_{0}^{1}fdx exists? - 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 why.
- Suppose f is a continuous function on and
where F is a strictly increasing integrator function. Show that then f

= 0 for all x. If 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 write

Explain.

- Let
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

⊆, then - Find ∑
_{k=1}^{∞}. - Let f be Riemann integrable on . Show directly that x →∫
_{0}^{x}fdt is continuous. Hint: It is always assumed that Riemann integrable functions are bounded. - 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 sense.
- Show
Now use this to find a series which converges to arctan

= π∕4. RecallFor 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≡∫
_{0}^{x}dt. Of course F= arctanas 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 lim_{x→∞}Fexists and show this limit equals −lim_{x→−∞}F. - In Problem 19 let the limit defined there be denoted by π∕2 and define T≡ F
^{−1}for x ∈. Show T^{′}= 1 + T^{2}and T= 0 . As part of this, you must explain why T^{′}exists. For x ∈let C≡ 1∕with C= 0 and on, define Sby. Show both Sand Care differentiable onand satisfy S^{′}= Cand C^{′}= −S. Find the appropriate way to define Sand Con all ofin order that these functions will be sinand cosand 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, [19] for this approach. - Show
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 = t

^{p−1}. Then solving for t, you get t = x^{1∕}= x^{q−1}. Explain this. Now let a,b ≥ 0. Sketch a picture to show whyNow 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 F, an increasing function. Verify Holder’s inequality.
Hint: Do the following. Let A =

^{1∕p},B =^{1∕q}. Then letand use the wonderful inequality of Problem 23.

- Let F= ∫
_{x2}^{x3 }dt. Find F^{′}. - Let F= ∫
_{2}^{x}dt. Sketch a graph of F and explain why it looks the way it does. - 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 ∈∫
fdx for all x ∈ ℝ, show F= G+ C for some constant, C. Use this to give a different proof of the fundamental theorem of calculus which has for its conclusion ∫
_{a}^{b}fdt = G− Gwhere G^{′}= f. - Suppose f is continuous on . Show there exists c ∈such that
Hint: You might consider the function F

≡∫_{a}^{x}fdt and use the mean value theorem for derivatives and the fundamental theorem of calculus. - Suppose f and g are continuous functions on and that g≠0 on. Show there exists c ∈such that
Hint: Define F

≡∫_{a}^{x}fgdt and let G≡∫_{a}^{x}gdt. 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 follows.
whenever this limit exists. Show

exists. Here the integrand is defined to equal 1 when x = 0, not that this matters.

- Show
exists.

- 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 fis an increasing function which is bounded above, then lim_{x→∞}fmust exist and equal sup. Then showis an increasing function of R which is bounded above.

- The most important of all differential equations is the first order linear equation,
y
^{′}+ py = fwhere p,f are continuous. Show the solution to the initial value problem consisting of this equation and the initial condition, y= y_{a}iswhere P

= ∫_{a}^{t}pds. Give conditions under which everything is correct. Hint: You use the integrating factor approach. Multiply both sides by e^{P(t) }, verify the left side equalsand then take the integral, ∫

_{a}^{t}of both sides. - Suppose f is a continuous function which is not equal to zero on . Show that
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, x
_{i−1},x_{i−1}+ h ≡ x_{i}, and x_{i}+ 2h ≡ x_{i+1}. Suppose also a function f, has the value f_{i−1}at x, f_{i}at x + h, and f_{i+1}at x + 2h. Then considerCheck that this is a second degree polynomial which equals the values f

_{i−1},f_{i}, and f_{i+1}at the points x_{i−1},x_{i}, and x_{i+1}respectively. The function g_{i}is an approximation to the function f on the interval. Also,is an approximation to ∫

_{xi−1}^{xi+1}fdx. Show ∫_{xi−1}^{xi+1}g_{i}dx equalsNow suppose n is even and

is a partition of the interval,and the values of a function f defined on this interval are f_{i}= f