ASSIGNMENT 2
There are two parts to this assignment. The first part consists of 15 questions on WeBWorK — the link is
available on the course webpage. The second part consists of the questions on this page. You are expected
to provide full solutions with complete justifications. You will be graded on the mathematical, logical and
grammatical coherence and elegance of your solutions. Your solutions must be typed, with your name and
student number at the top of the first page. If your solutions are on multiple pages, the pages must be stapled
together.
Your written assignment must be handed in before the start of your recitation on Friday, January
23. The online assignment will close at 9:00 a.m. on Friday, January 23.
1. Calculate lim R x2
x→0
x
x
1/3
(27 − 2t3 )
.
dt
2. A function f (x) is continuously differentiable if it is differentiable and its derivative is continuous. Find
all continuously differentiable functions f (x) such that
Z x
f (x)2 =
f (t)2 + f 0 (t)2 dt + 1.
0
3. Recall the Mean Value Theorem for integrals (MVTI): if f (t) is continuous on [a, b], then there exists a
number c in [a, b] such that
Z b
f (c)(b − a) =
f (t) dt.
a
In class, we proved this theorem using the Extreme Value Theorem (EVT) and the Intermediate Value
Theorem (IVT), and then used it to prove the Fundamental Theorem of Calculus (FTC). This particular
approach can be illustrated as follows:
EVT, IVT =⇒ MVTI =⇒ FTC.
However, the Fundamental Theorem of Calculus can be proved without appealing to the Mean Value
Theorem for integrals. (The sketch of such a proof is available on the course webpage.)
(a) Prove the Mean Value Theorem for integrals in a different way than shown in class, by using the
Fundamental
Theorem of Calculus. (Hint: apply the Mean Value Theorem (MVT) to the function
Rx
f
(t)
dt,
and
aim to get the following path:
a
MVT, FTC =⇒ MVTI.)
(b) Compare the following two proofs of the Mean Value Theorem for Integrals: your proof in part (a),
and the proof presented in class. Which is more attractive? Defend your answer in a few sentences.