- Let F =
dt. Find F′
- Let F =
dt. Sketch a graph of F and explain why it looks the way
- 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 conclusion ∫
dt = G
− G where
- Suppose f is Riemann integrable on and continuous. (In fact continuous
implies Riemann integrable.) 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? Explain why or why not.
- Prove the second part of Theorem 3.3.2 about decreasing functions.
- Suppose f is a bounded function defined on and
< M for all
. Now let Q be a partition having n points, and let
P be any
other partition. Show that
Hint: Write the sum for U
− L and split this sum into two sums, the
sum of terms for which
contains at least one point of
Q, and terms for
which does not contain any points of
Q. In the latter case, must
be contained in some interval,
. Therefore, the sum of these terms should
be no larger than
- ↑ If ε > 0 is given and f is a Darboux integrable function defined on , show
δ > 0 such that whenever
< δ, then
- ↑ Prove Theorem 3.5.6.