Mathematics 242 – Principles of Analysis Problem Set 9 – Due: Friday, April 19

‘A‘ Section

1. Let f(x) = x^{2} + 3x + 3 on [1, 2].

(a) Show that f is integrable on [1, 2] directly using the definition (that is, do not use any general theorems giving criteria for integrability). Hints: Use regular partitions of [1, 2], and the summation rules

∑_{i=1}^{n} i = \frac{n(n + 1)}{2} and ∑_{i=1}^{n} i^{2} = \frac{n(n + 1)(2n + 1)}{6}.

(b) Determine the value of ∫_{0}^{2} x^{2} + 3x + 3 dx.

2. Explain why the following inequalities must be true without evaluating the integrals involved:

(a)

0 < ∫_{0}^{π/2} \frac{sin(x) + cos(x)}{x^{3} + 1} dx < \frac{π}{\sqrt{2}}

(b)

\frac{1}{2} < ∫_{0}^{1} \frac{1 + x − x^{2}}{1 + tan (\frac{πx}{4})} dx < \frac{5}{4}

3. Let F (x) = ∫_{0}^{x} f(t) dt where

f(t) = \{ \begin{matrix}1 & if 0 ≤ t ≤ 1 \\ t + 1 & if 1 < t ≤ 2 \end{matrix}

(a) Find an explicit formula for F (x) valid for all 0 ≤ x ≤ 2.

(b) Is F differentiable at x = 1? Why or why not? Does this contradict the first part of the FTC?

4. Find the following derivatives using the FTC:

(a) \frac{d}{dx} ∫_{0}^{x} \frac{sin(t)}{t} dt

(b) \frac{d}{dx} ∫_{−x^{2}}^{x^{3}} e^{−u^{2}} du ‘B‘ Section

1. Let f be integrable on the interval [a, b], and assume ∫_{a}^{b} f(x) dx > 0. Show that there exist k > 0 and an interval [c, d] ⊆ [a, b] such that f(x) > k > 0 for all x ∈ [c, d].

2. Let f be continuous on [a, b]. Show that there exists c ∈ [a, b] such that

∫_{a}^{b} f(x) dx = f(c)(b − a).

(Hint: Look at Theorem 5.2.5 b.)

3. Logarithm and Exponential. In this problem, we will construct the natural logarithm and and the exponential function e^{x} “from scratch,” without relying on intuition about exponentials (as you probably did in calculus). We start by considering the function

(1)  L(x) = ∫_{1}^{x} \frac{1}{t} dt,

for x > 0. Note that \frac{1}{t} is continuous on (0, +∞), hence the FTC applies to show that L is a differentiable function. From calculus you probably recognize that L(x) = ln(x). We want to show directly that this makes sense and use this function to construct the inverse function E(x) = L^{−1}(x) which is callled E(x) = e^{x}. Why would we proceed “backwards” like this? The issue is that, while a^{m/n} = (a^{1/n})^{m} makes immediate sense for any positive a ∈ R and any rational exponent, what does a^{x} actually mean if x ∉ Q? Instead of trying to define that directly, we will take an end run around the question.

(a) We claim that the function L “has the right property to be a logarithm” – namely that L(x · x′) = L(x) + L(x′) for all x, x′ > 0. Prove this by showing that for any constant x′ > 0, the function L(xx′) also satisfies \frac{d}{dx}L(xx′) = \frac{1}{x} for all x > 0. Deduce that L(xx′) = L(x) + c for some constant c, then determine c by substituting an appropriate value for x.

(b) Show that L(x) is strictly increasing for x > 0, hence is a 1-1 function on the domain (0, +∞). Hence L has an inverse function E : R → {x ∈ R | x > 0}.

(c) Show that the inverse function E satisfies the equation E(x+x′) = E(x)·E(x′), hence E looks like an exponential function.

(d) We define x = e as the unique solution of the equation L(x) = 1. Show that x = e^{m/n} satisfies L(x) = m/n for all rational numbers x = m/n.

(e) Show that the inverse function E(x) of L(x) is differentiable and satisfies E′(x) = E(x). Hint: Look at Section 4.4. What happens if you differentiate on both sides of the equation L(E(x)) = x?

(f) Show that

E(x) = e^{x} = lim_{n→∞} (1 + \frac{x}{n})^{n}

for all x ∈ R. Hint: Take logarithm and use L’Hopital’s Rule.